Tanmoy Rana

Saravanan Mani

Singh Dharm

Meena

Sunita Choudhary

Marie-Christine Cadiergues
email: marie-christine.cadiergues@envt.fr

Alice Lecru

Daniel Combarros

Eloy Castilla-Castaño

Maxence Delverdier

Charline Pressanti

Positive Outcome of a Suspected Drug-Associated (Immune Mediated) Reaction in a 4-Year-Old Male French Bulldog

Keywords: beta-lactams antibiotic, dog, drug reaction, skin, toxic epidermal necrolysis

Toxic epidermal necrolysis (TEN) is a rare and severe life-threatening syndrome characterized by apoptosis of keratinocytes resulting in devitalization of the epidermis affecting more than 30% of skin surface. In humans and animals, this condition is mostly triggered by drugs. Identification of the putative agent and its withdrawal are crucial to successful management of a patient with TEN. In this case study, we report the clinical features, histopathological findings and management of a dog with TEN. A 4-year-old intact male French bulldog presented with acute onset of severe lethargy and cutaneous ulcerations on the footpads, scrotum, and hind limbs associated with marked pain. A Stevens-Johnson syndrome/TEN was suspected and drugs, especially beta-lactams, were withdrawn. Histopathology confirmed the diagnosis of epidermal necrosis. Advanced supportive therapy, pain management and skin care led to rapid remission. Early identification and removal of the suspected medication was crucial to improving TEN prognosis in this dog. Antibiotics (penicillin, ampicillin, cephalexin, and sulfonamides) are frequently involved in adverse cutaneous reactions in dogs. Ideal treatment remains elusive is humans and dogs and this disease has a poor prognosis. Supportive care combined with pain management and treatment of the cutaneous ulcerations is essential.

INTRODUCTION

Toxic epidermal necrolysis (TEN) is a rare and severe life-threatening syndrome characterized by apoptosis of keratinocytes of all epidermal layers resulting in devitalization of the epidermis in more than 30% of the skin surface [START_REF] Harris | Review of toxic epidermal necrolysis[END_REF]. After a phase of fever and malaise, TEN ultimately results in extensive skin involvement with erythema, necrosis, and detachment of the epidermis and mucosa. The individuals are then at high risk of developing sepsis and toxic shock [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF][START_REF] Gross | Skin diseases of the Dog and Cat: Clinical and Histopathologic Diagnosis[END_REF]. In humans and animals, TEN, and Stevens Johnson syndrome (SJS) are mostly triggered by drugs [START_REF] Mason | Cutaneous drug eruptions with epidermal necrosis: a discussion of pathophysiologic and comparative aspects[END_REF][START_REF] Voie | Drug hypersensitivity reactions targeting the skin in dogs and cats[END_REF] and differ from the surface of the epidermal detachement. Identification of the putative agent and its withdrawal are crucial to successful management of a patient with SJS and TEN in particular [START_REF] Fernando | The management of toxic epidermal necrolysis[END_REF][START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF]. We aim to report clinical features, histopathological findings and positive outcome of a suspected drug-associated (immune mediated) reaction in a 4-year-old male French bulldog with a dramatic initial presentation.

CASE DESCRIPTION

A 4-year-old intact male French bulldog presented with a 1week history of lethargy, dysorexia and painful erythematous macules and patches, the epidermis on the scrotum, perianal region, footpads, and hind limbs being readily detachable. One week earlier, a first veterinarian had treated the dog for superficial injuries (erosions, crusts and traumatic linear ulcers) secondary to a road traffic accident with three consecutive subcutaneous injections of amoxicillin [15 mg/kg (15-20 mg/kg)-Clamoxyl suspension R , Zoetis, Malakoff, France] and one subcutaneous injection of steroidal anti-inflammatory [dexamethasone, 0.1 mg/kg (0.1-0.2 mg/kg)-Dexadreson R , Intervet, Beaucouze, France]. A second veterinarian was consulted when new cutaneous lesions (multiple erosive, crusted and circular between 5 and 10 mm in diameter) appeared spontaneously on the ventral abdomen a week later. The dog was then treated with allopurinol [400 mg once daily, (10 mg/kg/twice a day)-Allopurinol Teva 200 mg, TEVA, Paris, France] after a presumptive diagnosis of leishmaniosis. A complete blood count (CBC) and serum biochemistry profile was obtained; the only abnormality identified was a moderate increase of alanine aminotransferase (ALT). The dog also received cephalexin [22.4 mg/kg twice daily (20-30 mg/kg)-Rilexine R 300 mg, Virbac, Carros, France], and a single intramuscular injection of furosemide [4 mg/kg (2.5-5 mg/kg)-Dimazon R , Intervet, Beaucouze, France]. Within a few days the dog's condition was even worse. Severe systemic signs, lethargy, anorexia, weight loss and pain were apparent and the cutaneous lesions rapidly developed into widespread ulcerations. Further subcutaneous injections of amoxicillin [15 mg/kg (15-20 mg/kg)-Clamoxyl suspension R , Zoetis, Malakoff, France] and dexamethasone [0.1 mg/kg (0.1-0.2 mg/kg)-Dexadreson R , Intervet, Beaucouze, France] did not improve the lesions or the general condition, and the dog was referred. He was presented with lethargy, respiratory distress, hyperthermia, and a 4-day history of anorexia.

On admission at the veterinary teaching hospital, general examination revealed a poor body condition score (2/9-8.6 kg), reluctance to walk, dyspnea and tachypnea, fever (39.8 • C) and enlargement of the prescapular lymph nodes. Severe extensive ulcerations were present bilaterally on the inner aspect of the hind legs (Figure 1A), between the digits (Figure 1B) and footpads (Figure 1C). The perianal region was eroded and depigmented (Figure 1D). A thick, adherent crusty plaque covered the scrotum which was almost completely ulcerated (Figure 1E). Large poorly-defined purpuric macules and patches were noted on the caudal aspect of the thigh (Figure 1F). The examination of these lesions was difficult because of the pain; the dog was moving and could not support the palpation. The pruritus was important and estimated 9/10 on a pruritus Visual Analog Scale by the owner. Major epidermal detachment (Figure 1G), very painful necrotizing lesions, and a pseudo-Nikolsky sign epidermal detachment obtained by gentle pressure to the erythematous skin, were apparent together with slight mucocutaneous depigmentation (Figure 1H). The authors estimated a surface of ∼1,500 cm² of extended lesions for a total body surface area (BSA) of 0.424 m² (dog's weight: 8.6). Hence, more than 35% of the BSA displayed ulcerations and epidermal sloughing.

The timeline of the case is shown in Figure 2.

The differential diagnosis included traumatic burns but was not supported by the history. Toxic epidermal necrolysis (TEN), erythema multiforme (EM) and vasculitis with ischemic necrosis were possible considering the acute onset and widespread nature of the ulcerations. Bacterial or viral infection, staphylococcal toxic shock syndrome and bullous auto-immune diseases (pemphigus and pemphigoid diseases, epidermolysis bullosa acquisita) could be included because of the erosive and ulcerative aspect of the lesions, the mucosal involvement and the systemic illness. Systemic lupus erythematosus could also be considered.

Cytological examination of impression smears from the limb ulcerations showed many neutrophils, eosinophils and intracytoplasmic coccoid bacteria (phagocytosis). Cutaneous biopsies were performed on the caudal part of the right forelimb and back thigh under general anesthesia using acepromazine, propofol and isoflurane. Four 6-mm biopsy punch samples were taken from the margins of ulcerative lesions and from the normal skin surrounding the lesions and stained with hemalun and eosin following standard procedures.

A cutaneous drug reaction was highly suspected. A specific investigation of all xenobiotics the dog was receiving at the time of the reaction was made with the owner. Beta-lactams were the single current therapy. Using the Naranjo Algorithm Adverse Drug Reaction Probability Scale (8), a score of 4 of 13 was obtained (Table 1). The causality was considered possible. Beta-lactams were withdrawn.

Histological examination revealed severe extensive multifocal epidermal necrosis with multifocal epidermal detachment and necrotic material within flaccid vesicles (Supplementary Figure 1A). Some extension of ulcerations to the hair follicles was noted (Supplementary Figure 1B). Minimal dermal inflammation was present. Slight heterogeneous perivascular dermatitis with lymphocytes, neutrophils and plasma cells was observed (Supplementary Figure 1C). Some mild multifocal septal panniculitis was present (Supplementary Figure 1D). Taking together the histological findings and the clinical appearance, the hypothesis of TEN was confirmed.

While awaiting the bacterial culture results, an initial intravenous treatment of marbofloxacin (Marbocyl R , Vetoquinol, Lure, France) at 4 mg/kg (2-4 mg/kg) was administered once daily to prevent sepsis. Still under general anesthesia, ulcer wound management was initiated which consisted of gentle shaving, delicate removal of detached epidermis and disinfection with concentrated chlorhexidine shampoo (Hibiscrub R , Regent medical overseas limited, Manchester, United Kingdom). Soft-adherent dressings with poly-absorbent fibers, UrgoClean R (Elevate SAS, Saint-Romainen-Viennois, France), which trapped the sloughed residues and absorbed any exudate, and UrgoTul Absorb R (Elevate SAS, Saint-Romain-en-Viennois, France), which stimulated fibroblast proliferation and ensured non-adherence, were changed every day. A high-protein diet was introduced (Royal Canin Obesity ND, Royal Canin, Aimargues, France) to support healing.

By day two, an improvement was already apparent, the skin had started healing and granulation tissue was observed (Supplementary Figure 2). The dog had a good appetite and 2. Did the adverse event appear after the suspected drug was administered?

+2 -1 0 2
Yes (Timeline schematic)

3. Did the adverse event improve when the drug was discontinued or a specific antagonist was administered? +1 0 0 1 Improvement after discontinuation and supportive care.

4. Did the adverse event reappear when the drug was readministered? +2 -1 0 0

The drug was not readministered.

5. Are there alternative causes that could on their own have caused the reaction?

-1 +2 0 -1
Allopurinol as a cause of TEN is described in human medicine [START_REF] Wang | Allopurinol-induced toxic epidermal necrolysis featuring almost 60% skin detachment[END_REF][START_REF] Buenrostro-Rubio | Allopurinol-induced toxic epidermal necrolysis[END_REF] 6. Did the reaction reappear when a placebo was given? -1 +1 0 0 No placebo was given. no digestive disorder or pain was noted. Dressings were changed under sedation every other day for 1 week. Expansion of the cutaneous lesions was halted and analgesics were gradually decreased. Marbofloxacin was maintained based on the sensitivity results received on day 15, 3 days after presentation (Figure 2). After 2 weeks, the footpads and digits had healed. Localized perionyxis (Supplementary Figure 3A), with dark crusts and suppurative oozing, was noted. Cytological examination of a cotton swab of this lesion showed numerous neutrophils, and intracytoplasmic coccoid bacteria. Bacterial culture revealed a strain of methicillin-resistant S. pseudintermedius, only sensitive to fusidic acid, tetracyclines and chloramphenicol. The antibiotherapy was modified and doxycycline (Doxyval R , CEVA, Libourne, France) was administered orally once daily at the recommended dose 10 mg/kg. In the absence of topical tetracycline availability in the country, fusidic acid ointment (Forudine R , Dechra, Uldum, Denmark) was applied to the skin underneath the dressings every other day, because the perionyxis was painful and handling had to be limited. Crusted granulation tissue was apparent on the caudal side of the distal limbs but no oozing was noted (Supplementary Figure 3B). Dressings were changed every 2 days by the veterinarian and the owner. The dog had gained weight (9.4 kg) and was in good general health. A moisturizing cream (Trixera+ R , Avène, Paris, France) was applied every day to facilitate skin healing. Oral doxycycline was maintained.

After 3 weeks from starting doxycycline, the improvement was excellent, the distal limbs had completely healed (Figure 3A) but some crusty and inflammatory lesions were observed on the back of the thigh (Figure 3B) and the extremities, particularly around the claws (Figure 3C). Onychomadesis was observed on digits II and IV of the left hind foot (Figure 3D). Cytological examination of impression smears from crusty lesions on the caudal part of the hind legs revealed many neutrophils and rare bacteria. Doxycycline was maintained. Antiseptic shampoos (Douxo Pyo R , CEVA, Libourne, France) were prescribed twice weekly. Moisturizing cream was still applied once daily.

After seven weeks, the dog's weight was 9.8 kg, the skin had completely healed and noticeable hair regrowth had occurred (Supplementary Figure 4). Doxycycline was discontinued and antiseptic shampoos were continued once a week.

On the follow-up examination, 2 months after the doxycycline discontinuation, the dog was in a good body condition, and hair regrowth was almost complete.

DISCUSSION

TEN, also called ≪Lyell syndrome≫ is a rare life-threatening disorder reported in people and animals [START_REF] Harris | Review of toxic epidermal necrolysis[END_REF][START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF]. The hallmark of this syndrome is acute onset of epidermal detachment over a significant surface with associated systemic signs. It must be treated as an emergency. The prognosis is extremely poor, due mainly to consecutive sepsis. In humans, there is a high mortality rate of 30% and considerable long-term morbidity [START_REF] Bastuji-Garin | a severity-of-illness score for toxic epidermal necrolysis[END_REF]. The mortality rate for TEN in animals, based on relatively few case reports, is very high [START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF].

Antibiotic drugs, especially sulfonamides, penicillins, and cephalosporins are most often incriminated [START_REF] Mason | Cutaneous drug eruptions with epidermal necrosis: a discussion of pathophysiologic and comparative aspects[END_REF][START_REF] Miller | Toxic epidermal necrosis[END_REF][START_REF] Scott | Immune-mediated dermatoses in domestic-animals -10 years after -part II[END_REF][START_REF] Noli | retrospective evaluation of adverse reactions to trimethoprim-sulphonamide combinations in dogs and cats[END_REF]. The pathomechanism is presumed to be immune-mediated, involving a cytotoxic CD8+ T-cell and natural killer (NK) cell activation against altered keratinocytes [START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF][START_REF] Banovic | Apoptosis as a mechanism for keratinocyte death in canine toxic epidermal necrolysis[END_REF]. Soluble mediators including Fas-ligand, tumor necrosis factor-α (TNF-α), perforin, granzym, and mainly granulysin lead to confluent cell death [START_REF] Harris | Review of toxic epidermal necrolysis[END_REF][START_REF] Chung | Granulysin is a key mediator for disseminated keratinocyte death in Stevens-Johnson syndrome and toxic epidermal necrolysis[END_REF].

SJS and TEN are related conditions, distinguished by the percentage of BSA affected: <10% of the BSA affected is indicative of SJS, 10-30% indicates an overlap between SJS and TEN, and more than 30% of the BSA affected suggests TEN [START_REF] Bastuji-Garin | Clinical classification of cases of toxic epidermal necrolysis, Stevens-Johnson syndrome, and erythema multiforme[END_REF].

TEN is characterized by an acute onset of multifocal to generalized purpuric macules or patches, frequently extending to the mucocutaneous regions [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF][START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF]. Some lesions rapidly progress to vesicular/bullous, necrotic and ulcerative lesions with detachable epidermis, and a positive pseudo-Nikolsky sign [START_REF] Bastuji-Garin | Clinical classification of cases of toxic epidermal necrolysis, Stevens-Johnson syndrome, and erythema multiforme[END_REF][START_REF] Paquet | New insights in toxic epidermal necrolysis (Lyell's syndrome): clinical considerations, pathobiology and targeted treatments revisited[END_REF]. Footpads and digits are often involved [START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF]. Systemic signs of illness (lethargy, depression, anorexia, and fever), severe pain and pruritus are common.

In this case, focal lesions of depigmentation were present on the upper lip and the perianal region was largely ulcerated. However, the oral cavity, which is a common localization [START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF], was not involved. Extension was very rapid with more than 30% of the BSA initially affected.

Differential diagnosis should include auto-immune subepidermal blistering diseases, accidental burns erythema multiforme, staphylococcal toxic shock syndrome and bacterial or viral infections [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF][START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF]. The animal's history and the physical examination are mandatory steps in the diagnosis. Other steps include questioning the owner about previous cutaneous reactions, any current treatments: antibiotics, nutraceuticals, deworming tablets, non-steroidal anti-inflammatory drugs, and any topical drug the dog could receive at the time of the reaction. Any previous exposure to the drug(s) suspected has to be investigated. Finally, the owner has to be questioned about the time between treatments and lesion development. When a drug reaction is suspected, any current xenobiotic has to be withdrawn immediately unless it represents an acutely life-threatening risk.

The Naranjo Scale was developed in human medicine to help standardize assessment of causality for all adverse drug reactions and establishes a causal association between a drug and an adverse event. It consists of 10 questions that are answered as either Yes, No, or "Do not know." Different point values (-1, 0, +1, or +2) are assigned to each answer [START_REF] Naranjo | A method for estimating the probability of adverse drug reactions[END_REF]. Total scores range from -4 to +13; the reaction is considered definite if the score is 9 or higher, probable if 5-8, possible if 1-4, and doubtful if 0 or less. Other scales are available [START_REF] Bastuji-Garin | a severity-of-illness score for toxic epidermal necrolysis[END_REF][START_REF] Sassolas | an algorithm for assessment of drug causality in Stevens-Johnson Syndrome and toxic epidermal necrolysis: comparison with case-control analysis[END_REF]. No equivalent exists for the dog, but their extrapolation is helpful.

For differential diagnostic purpose and suspecting TEN consequences, thoracic radiographs and abdominal ultrasound scans were obtained to rule out other organ damage secondary to possible sepsis or staphylococcal toxic shock syndrome [START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF][START_REF] Scott | Toxic epidermal necrolysis in two dogs and a cat[END_REF].

Interface dermatitis with extensive full-thickness epidermal necrosis and minor dermal inflammation are characteristic histological features although not pathognomonic for SJS/TEN [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF][START_REF] Gross | Skin diseases of the Dog and Cat: Clinical and Histopathologic Diagnosis[END_REF][START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF][START_REF] Banovic | Apoptosis as a mechanism for keratinocyte death in canine toxic epidermal necrolysis[END_REF]. However, in some occasions, this feature is not observed but rather a cytotoxic dermatitis with apoptosis of individual keratinocytes and satellitosis in multiples layers of the epidermis, signs that overlap histologically with erythema multiforme (EM) [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF]. In these cases, the clinician plays a key role in distinguishing EM from SJS/TEN. Multiple skin biopsies are recommended to confirm the diagnosis as it was shown that up to 25% of all skin biopsies lacked epithelium and thus were not diagnostic [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF]. Progressive lesions, ulcer margins, or the adjacent intact epidermis are sampling sites of interest [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF][START_REF] Gross | Skin diseases of the Dog and Cat: Clinical and Histopathologic Diagnosis[END_REF].

In our case, the dog arrived in a state of shock, dehydrated, and in extreme pain. Our first steps were to manage the pain, replace fluids and electrolytes, and prevent sepsis. As a drug reaction caused by amoxicillin was suspected, penicillins were withdrawn and replaced by antibiotics that have been less commonly associated with adverse cutaneous drug reactions (Supplementary Table 1). Constant patient monitoring, skin care and pain management were maintained for 15 days. Response to treatment was prompt and the dog improved the very next day. This improvement suggested that betalactams could be the initial trigger of the cutaneous reaction; furthermore, despite the use of marbofloxacin (resistant bacterial profile), the dog's condition improved, emphasizing the importance of the intensive care in the management.

According to the literature, adjunctive treatments have been used to counteract the immune mechanism behind SJS and TEN but in human medicine no survival benefits have been proved for glucocorticoids or high doses of human intravenous immunoglobulin (IV Ig) [START_REF] Harris | Review of toxic epidermal necrolysis[END_REF][START_REF] Creamer | UK guidelines for the management of Stevens-Johnson syndrome/toxic epidermal necrolysis in adults 2016[END_REF]. Ciclosporin A (CsA) has been associated with increased survival benefits in humans in many case reports and uncontrolled small studies [START_REF] Reese | Cyclosporine for SJS/TEN: a case series and review of the literature[END_REF][START_REF] Valeyrie-Allanore | Open trial of ciclosporin treatment for Stevens-Johnson syndrome and toxic epidermal necrolysis[END_REF]. Its effects may be attributed to its ability to inhibit cytotoxic T cells and its action on the apoptotic pathway which precludes the release of cytotoxic mediators [START_REF] Arevalo | Treatment of toxic epidermal necrolysis with cyclosporin A[END_REF]. However, one study showed no significant difference in survival among all three treatment options of the study (CsA, IV Ig, supportive care) [START_REF] Firoz | Toxic epidermal necrolysis: five years of treatment experience from a burn unit[END_REF]. A doubleblinded controlled study is now required in veterinary medicine to determine the ideal dose and duration of treatment [START_REF] Fernando | The management of toxic epidermal necrolysis[END_REF]. In veterinary medicine, the cost of such treatment could be rapidly a brake, in particular when treating large breeds. Currently, the ideal immunomodulatory drug for the treatment of SJS/TEN remains elusive [START_REF] Fernando | The management of toxic epidermal necrolysis[END_REF][START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF][START_REF] French | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis[END_REF].

It is difficult to identify a candidate drug because provocation testing is not ethically acceptable, and because there is no fully validated in vitro/ex vivo/ in vivo models of drug allergy. In the present case, the use of the algorithms of drug causality for epidermal necrolysis (ALDEN) [START_REF] Sassolas | an algorithm for assessment of drug causality in Stevens-Johnson Syndrome and toxic epidermal necrolysis: comparison with case-control analysis[END_REF] and Naranjo scale (8) gave a score of 5 and 4 for amoxicillin, respectively, which corresponds to a probable causality. These results should be interpreted with caution as these algorithms have been developed for humans and have not been validated in dogs. Early withdrawal of all the xenobiotics the animal is exposed to at the time (only exception: acute life-threatening risk associated with the abrupt withdrawal molecule), referral to a specialized center, and optimal supportive treatment are crucial to the successful management of patients [START_REF] Banovic | Clinical and microscopic characteristics of canine toxic epidermal necrolysis[END_REF][START_REF] Voie | Drug hypersensitivity reactions targeting the skin in dogs and cats[END_REF][START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF]. In humans, identifying the causative drug can reduce the chance of death in TEN patients by 30% per day [START_REF] Yager | Erythema multiforme, Stevens-Johnson syndrome and toxic epidermal necrolysis: a comparative review[END_REF]. TEN usually has a poor prognosis; the mortality rate in humans is 25-35% despite early diagnosis and relative easy access to intensive care units or burn units [START_REF] Harris | Review of toxic epidermal necrolysis[END_REF].

CONCLUSION

In the present case, the good improvement of the dog's condition and the cutaneous lesions suggest that betalactams were the culprit drugs. As fluoroquinolones have never been reported to produce cutaneous drug reactions in dogs and cats, marbofloxacin was administered to prevent sepsis. A methicillin-resistant Staphylococcus pseudintermedius sensitive to fusidic acid, tetracyclines, chloramphenicol and resistant to marbofloxacin was isolated from the bacterial cultures. Doxycycline was prescribed, wound management using fusidic acid ointment and antiseptic treatments were maintained. The observed skin improvement and good general condition, despite the use of marbofloxacin was probably due to the intensive care and pain management. The present case illustrates the importance of rapid identification of any acute onset of necrolysis and epidermal detachment in animals. Drug treatments are most often involved and should automatically be stopped.

FIGURE 1 |

 1 FIGURE 1 | Initial physical examination of a 4-year-old intact male French bulldog. Severe ulcerations involving the inner aspect of the hind legs (A) and ulcerations of the interdigital region (B) and the footpads (C). Ulcerations, erosions and depigmentation of the perianal region (D) and thick crusty material covering the scrotum (E). Purpuric macules and patches were present on the caudal aspects of the thighs (F). Epidermis was sloughing off around the ulcerative lesions (G). Slight depigmentation was visible on the upper lip (H).

FIGURE 2 |

 2 FIGURE 2 | Schematic diagram describing timeline clinical, diagnostic and treatments features of suspected drug-associated (immune mediated) reaction in a 4-year-old male French bulldog. SPI, Staphylococcus pseudintermedius.

  A sample was obtained by cotton swab for bacterial culture. Strains of Staphylococcus pseudintermedius sensitive to all the antibiotics tested, including betalactams and fluoroquinolones, were isolated. CBC revealed severe neutrophilic leukocytosis (16.10 9 /L). Serum biochemistry was unremarkable except for a huge increase of fibrinogen [8.87 g/L (1.3-4.8)], moderate increase of alkaline phosphatases [346 IU/L (23-212)] and minor hypochloremia [103 mmol/L (110-118)]. Prothrombin time and activated partial thromboplastin time were within the normal ranges. Urinalysis highlighted numerous struvite crystals without proteinuria (urine protein to creatinine ratio = 0.4). Abdominal ultrasonography and thoracic radiographs did not reveal any abnormality. An indirect immunofluorescence test [Enzymelinked immunosorbent assay (ELISA)] for leishmaniasis was negative. A canine lipase specific SNAP test (ELISA) was normal. Fine-needle aspiration of the enlarged lymph nodes revealed granulomatous adenitis associated with plasma cell hyperplasia.

7 . 8 . 9 .

 789 Was the drug detected in blood or other fluids in concentrations known to be toxic? Was the reaction more severe when the dose was increased or less severe when the dose was decreased? Did the patient have a similar reaction to the same or similar drugs in any previous exposure? No similar reaction previously.10. Was the adverse event confirmed by any objective evidence?

FIGURE 3 |

 3 FIGURE 3 | Clinical examination after 3 weeks. Healing was complete along the distal limbs (A), crusty inflammatory lesions were observed on the back of the left thigh (B) and the extremities, particularly around the claws (C). Onychomadesis was observed on digits II and IV of the left hind foot (D).

TABLE 1 |

 1 Naranjo adverse drug reaction probability scale applied to the case study.

	Question

Frontiers in Veterinary Science | www.frontiersin.org

August 2021 | Volume 8 | Article 728901

ACKNOWLEDGMENTS

The authors thank Dr. F. Lefournier who referred the case to the VTH. They are very grateful to the owner of the dog for her continuing collaboration. They thank Diana Warwick for English editing. They would like to acknowledge Vetoquinol and Lilly Elanco for supporting the positions of L-AL and EC-C and as ECVD residents, respectively.

DATA AVAILABILITY STATEMENT

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

ETHICS STATEMENT

Ethical review and approval were not required as no original research data was generated in this case report. All diagnostic and therapeutic procedures were performed by licensed veterinarians in the course of routine veterinary health management. Written informed consent was obtained from the owners for the participation of their animals in this study.

AUTHOR CONTRIBUTIONS

L-AL: investigation, follow-up, original draft, and review and editing. DC, EC-C, and M-CC: investigation and review and editing. MD: investigation (histopathology) and review and editing. CP: investigation, follow-up, and review and editing. All authors contributed to the article and approved the submitted version.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fvets. 2021.728901/full#supplementary-material

Conflict of Interest:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Copyright © 2021 Lecru, Combarros, Castilla-Castaño, Delverdier, Cadiergues and Pressanti. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.


Rémi Carles
email: remi.carles@math.cnrs.fr

Guillaume Ferriere
email: guillaume.ferriere@math.unistra.fr

Logarithmic Schrödinger equation with quadratic potential

We analyze dynamical properties of the logarithmic Schrödinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.

Introduction

We consider the logarithmic Schrödinger equation in the presence of an external potential, (1.1) i∂ t u + 1 2 ∆u = V (x)u + λ ln |u| 2 u, u |t=0 = u 0 , with x ∈ R d , d 1, and λ ∈ R. The potential V is smooth and real-valued, V ∈ C ∞ (R d ; R), and the main results of the present paper concern the case where V is quadratic in x. This equation is Hamiltonian; mass and energy are formally independent of time:

(1.2)

M (u(t)) = u(t) 2 L 2 (R d ) , E(u(t)) := 1 2 ∇u(t) 2 L 2 (R d ) + R d V (x)|u(t, x)| 2 dx + λ R d |u(t, x)| 2 ln |u(t, x)| 2 -1 dx.
On the other hand, due to the presence of the potential V , the momentum,

Im R d ū(t, x)∇u(t, x)dx,
is not conserved. As it will not be considered in this paper, we do not discuss the evolution of this quantity. The logarithmic nonlinearity was introduced in [START_REF] Białynicki-Birula | Nonlinear wave mechanics[END_REF] to satisfy the following RC is supported by Rennes Métropole through its AIS program.

1 tensorization property: if V decouples space variables in the sense that

V (x) = d j=1 V j (x j ),
where V j ∈ C ∞ (R; R), and if the initial datum is a tensor product,

u 0 (x) = d j=1 u 0j (x j ),
then the solution to (1.1) is given by

u(t, x) = d j=1 u j (t, x j ),
where each u j solves a one-dimensional equation,

i∂ t u j + 1 2 ∂ 2 xj u j = V j (x j )u j + λ ln |u j | 2 u j , u j|t=0 = u 0j .
Of course, a similar property remains if e.g. V (x 1 , x 2 , x 3 ) = V 12 (x 1 , x 2 ) + V 3 (x 3 ), and u 0 (x 1 , x 2 , x 3 ) = u 012 (x 1 , x 2 )u 03 (x 3 ). As can easily be checked, the above logarithmic nonlinearity is the only one satisfying this tensorization property. This property will be evoked several times in the rest of this paper.

The logarithmic nonlinearity has since been adopted in several physical models, in quantum mechanics [START_REF] Yasue | Quantum mechanics of nonconservative systems[END_REF], quantum optics [START_REF] Białynicki-Birula | Nonlinear wave mechanics[END_REF][START_REF] Hansson | Propagation of partially coherent solitons in saturable logarithmic media: A comparative analysis[END_REF][START_REF] Krolikowski | Unified model for partially coherent solitons in logarithmically nonlinear media[END_REF][START_REF] Buljan | Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media[END_REF], nuclear physics [START_REF] Hefter | Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics[END_REF], Bohmian mechanics [START_REF] Martino | Logarithmic Schrödinger-like equation as a model for magma transport[END_REF], effective quantum gravity [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF], theory of superfluidity and Bose-Einstein condensation [START_REF] Avdeenkov | Quantum Bose liquids with logarithmic nonlinearity: Selfsustainability and emergence of spatial extent[END_REF]. See also [START_REF] Scott | Solution of the logarithmic Schrödinger equation with a Coulomb potential[END_REF][START_REF] Shertzer | Solution of the 3D logarithmic Schrödinger equation with a central potential[END_REF]. The papers [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF][START_REF] Zloshchastiev | Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory[END_REF][START_REF] Bouharia | Stability of logarithmic Bose-Einstein condensate in harmonic trap[END_REF] have provided evidences that the logarithmic model may generalize the Gross-Pitaevskii equation, used in the case of two-body interaction, to the case of three-body interaction.

As proposed in [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF][START_REF] Zloshchastiev | Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory[END_REF], the logarithmic nonlinearity may appear as a relevant model to provide a universal mechanism describing the deformation of the quantum wave equation due to non-trivial vacuum. It thus appears as a serious candidate to extend quantum mechanics thanks to a nonlinear model, likely to help understand quantum gravity. In the absence of external potential (V = 0), the case λ < 0 is certainly the most physically relevant: stationary solutions known as Gaussons are present and stable under the dynamics (see below), while for λ > 0, (enhanced) dispersion is always present; see Theorem 1.1 for a complete mathematical statement, showing that it is indeed hopeless to look for stable structures if λ > 0.

In [START_REF] Bouharia | Stability of logarithmic Bose-Einstein condensate in harmonic trap[END_REF], the presence of an harmonic trap was considered, in order to describe logarithmic BEC. As in the case without potential, an important feature of the model is that many Gaussian solutions are available, as explored into more details in Section 3.3. Stationary solutions (generalized Gaussons) are still available in the case λ < 0. Their stability was investigated numerically in [START_REF] Bouharia | Stability of logarithmic Bose-Einstein condensate in harmonic trap[END_REF], and analyzed mathematically in [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF]. The presence of the harmonic trap implies that stable structures are now also present in the case λ > 0, as shown numerically in [START_REF] Bouharia | Stability of logarithmic Bose-Einstein condensate in harmonic trap[END_REF], and analyzed more precisely in the present paper.

Considering for V a harmonic potential is not only physically relevant, it has also a natural mathematical motivation, which we now explain.

1.1. Mathematical background. The first mathematical study of (1.1) appears in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] (see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]), where the Cauchy problem is addressed for V ∈ L p (R d ) + L ∞ (R d ), with p 1 and p > d/2. As noticed in [START_REF] Białynicki-Birula | Gaussons: Solitons of the logarithmic Schrödinger equation[END_REF], in the case λ < 0 with V = 0, (1.1) admits a solitary wave of the form e iωt φ(x), where φ is a Gaussian, called Gausson. The orbital stability of this solitary wave was established in [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] in the radial case, and in [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF] in the general case. Still in the case λ < 0 with V = 0, it was shown in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] that if u 0 ∈ W , the energy space given by

W = f ∈ H 1 (R d ), |f | 2 ln |f | 2 ∈ L 1 (R d ) ,
then there exists a unique, global, solution u ∈ L ∞ (R; W ) (see also [START_REF] Guerrero | Global solvability of the 3d logarithmic schrödinger equation[END_REF][START_REF] Hayashi | A note on the nonlinear Schrödinger equation in a general domain[END_REF]), and in [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] that no such solution is dispersive. The global Cauchy problem in the case V = 0, λ > 0 was treated in [START_REF] Guerrero | Global solvability of the 3d logarithmic schrödinger equation[END_REF] and [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF].

As noticed in [START_REF] Białynicki-Birula | Nonlinear wave mechanics[END_REF], in the case V = 0, λ ∈ R, if u 0 is a Gaussian, then u(t, •) is a Gaussian for all time. The evolution of Gaussians was analyzed more precisely in [START_REF] Bao | Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation[END_REF][START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF][START_REF] Ferriere | The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superposition[END_REF]: in the case λ < 0, they behave like breathers, in the sense that |u(t, •)| is periodic in time, while if λ > 0, they all are dispersive with a similar rate τ (t), and τ (t) d |u(t, xτ (t))| 2 has a universal limit, that is, a Gaussian whose variance does not depend on the initial Gaussian u 0 .

The latter property has been established for initial data which are not necessarily Gaussian, in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF], and the description of the convergence was refined in [START_REF] Ferriere | Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation[END_REF]. Denote

Σ = H 1 ∩ F (H 1 ) = f ∈ H 1 (R d ), x → |x|f (x) ∈ L 2 (R d ) . THEOREM 1.1 ([16, 22]). Let V = 0 and λ > 0. For u 0 ∈ Σ \ {0}, (1.1) has a unique solution u ∈ L ∞ loc (R; Σ)). Introduce the solution τ ∈ C ∞ (R) to the ODE (1.3) τ = 2λ τ , τ (0) = 1 , τ (0) = 0 .
Then, as t → ∞, τ (t) ∼ 2t √ λ ln t and τ (t) ∼ 2 √ λ ln t. Introduce γ(x) := e -|x| 2 /2 , and rescale the solution to v = v(t, y) by setting

(1.4) u(t, x) = 1 τ (t) d/2 v t, x τ (t) u 0 L 2 (R d ) γ L 2 (R d ) exp i τ (t) τ (t) |x| 2 2 .
Then we have

R d   1 y |y| 2   |v(t, y)| 2 dy -→ t→∞ R d   1 y |y| 2   γ 2 (y)dy, and |v(t, •)| 2 ⇀ t→∞ γ 2 weakly in L 1 (R d ).
Finally, denoting by W 1 the Wasserstein distance 1 , there exists C such that

W 1 |v(t)| 2 π d/2 , γ 2 π d/2 C √ ln t , t e.
1 For ν 1 and ν 2 probability measures,

W 1 (ν 1 , ν 2 ) = inf R d ×R d |x -y|dµ(x, y); (π j ) ♯ µ = ν j ,
where µ varies among all probability measures on R d × R d , and π j : R d × R d → R d denotes the canonical projection onto the j-th factor. See e.g. [START_REF] Villani | Topics in optimal transportation[END_REF] The case where V is harmonic, V (x) = |x| 2 , was considered in [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF], in the case λ < 0: there exists an analogue of the Gausson, that is a solution of the form b 0 e iωt e -a0|x| 2 /2 , and it is orbitally stable. In the following statement, we adapt the numerical values to (1.1) and unify the statements from [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF] (without potential) and [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF] (with potential, where (1.1) is no longer translation invariant): THEOREM 1.2 ( [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF][START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF]). Let d 1 and λ < 0. Suppose that

V (x) = κ(κ + 2λ) 2 |x| 2 , κ -2λ > 0.
Then the (generalized) Gausson is given by φ ν (x) = e -ν+κd/2 2λ e -κ|x| 2 /2 , for ν ∈ R. It generates a standing wave u(t, x) = φ ν (x)e iνt solution to (1.1), which is orbitally stable in the energy space: For any ε > 0, there exists η > 0 such that if u 0 ∈ X satisfies u 0 -φ ν X < η, then the solution u to (1.1) exists for all t ∈ R, and

• Case without potential: κ = -2λ and X = W ,

sup t∈R inf θ∈R inf y∈R d u(t) -e iθ φ ν (• -y) W < ε.
• Case with potential: κ > -2λ and X = Σ,

sup t∈R inf θ∈R u(t) -e iθ φ ν Σ < ε.

Main results.

To make things clear, we first show that the Cauchy problem (1.1) is globally well-posed in Σ, provided that the potential V is smooth and at most quadratic. This naturally generalizes the results known in the case of a power nonlinearity (see e.g. [START_REF] Carles | Nonlinear Schrödinger equation with time dependent potential[END_REF]). (1.1). Moreover, the mass M (u(t)) and the energy E(u(t)) are independent of time.

PROPOSITION 1.3. Let V ∈ C ∞ (R d ; R), at most quadratic, in the sense that ∂ α V ∈ L ∞ (R d ) as soon as |α| 2. For u 0 ∈ Σ, there exists a unique solution u ∈ L ∞ loc (R; Σ) ∩ C(R; L 2 (R d )) to
The above result was established in [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF] in the case where V is an isotropic harmonic potential, and λ < 0. We prove Proposition 1.3 by adapting the strategy from [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF], which is different from the one in [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF], inspired by [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF]. REMARK 1.4. The case of a time dependent potential V could be considered as well, with very few modifications regarding Proposition 1.3 (see [START_REF] Carles | Nonlinear Schrödinger equation with time dependent potential[END_REF]), as well as in the description of the propagation of Gaussian functions in Section 3.3.

In the rest of this introduction, and for the other results, we assume λ > 0. 1.2.1. Full harmonic confinement. We first consider the case of a full, isotropic confinement: V (x) = ω 2 2 |x| 2 . This confinement completely alters the dynamics of the case λ > 0, since solitary waves now exist, while from Theorem 1.1, all Σ-solutions are dispersive when V = 0. To make the connexion with the case λ < 0 considered in [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF] explicit, rewrite ω 2 as ω 2 = κ(κ + 2λ), κ > 0.

THEOREM 1.5. Let d 1 and λ > 0. Suppose that

V (x) = κ(κ + 2λ) 2 |x| 2 , κ > 0.
Then the (generalized) Gausson is given by φ ν (x) = e -ν+κd/2 2λ e -κ|x| 2 /2 , for ν ∈ R. It generates a standing wave u(t, x) = φ ν (x)e iνt solution to (1.1), which is orbitally stable in the energy space: For any ε > 0, there exists η > 0 such that if u 0 ∈ Σ satisfies u 0 -φ ν Σ < η, then the solution u to (1.1) exists for all t ∈ R, and

sup t∈R inf θ∈R u(t) -e iθ φ ν Σ < ε.
We see in particular that in the limit case V = 0, corresponding to κ = 0, φ ν is no longer an L 2 -function. To prove Theorem 1.5, we essentially resume the approach from [START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF], based on the Cazenave-Lions method [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF], as well as on a variational characterization of the generalized Gausson. In the case λ < 0, this characterization relies on the logarithmic Sobolev inequality and the description of equality cases. In the present framework, the logarithmic Sobolev inequality is replaced by a rather natural counterpart, involving a momentum instead of a derivative, see Lemma 4.2.

REMARK 1.6. The case of anisotropic confinement in all directions,

V (x) = d j=1 ω 2 j 2
x 2 j , ω j > 0, can be addressed with straightforward adaptations, by considering suitable anisotropic Gaussian functions in all steps of the proof, including Lemma 4.2.

1.2.2. Partial harmonic confinement. In the same spirit as e.g. [START_REF] Antonelli | Scattering for nonlinear Schrödinger equation under partial harmonic confinement[END_REF][START_REF] Bellazzini | Existence and stability of standing waves for supercritical NLS with a partial confinement[END_REF], we now assume d 2, and that V is confining in some but not all directions: suppose that the space variable is now

(x ′ , x ′′ ) ∈ R d1 × R d2 , with d 1 , d 2 1, d 1 + d 2 = d, and V (x ′ , x ′′ ) = ω 2 2 |x ′ | 2 .
We show that dispersion is always present in x ′′ , in the sense that Theorem 1.1 remains valid for u, provided that suitable integration in x ′ is considered.

THEOREM 1.7. Let (x ′ , x ′′ ) ∈ R d1 × R d2 , with d 1 , d 2 1, d 1 + d 2 = d, and
V (x ′ , x ′′ ) = ω 2 2 |x ′ | 2 , ω > 0.
Suppose λ > 0. Let u 0 ∈ Σ \ {0}, and resume the notation γ(x ′′ ) := e -|x ′′ | 2 /2 . Introduce

ρ(t, y) := τ (t) d2 R d 1 |u (t, x ′ , yτ (t))| 2 dx ′ × π d2 u 0 2 L 2 (R d )
.

Then we have

R d 2   1 y |y| 2   ρ(t, y)dy -→ t→∞ R d 2   1 y |y| 2   γ 2 (y)dy, and ρ(t, •) ⇀ t→∞ γ 2 weakly in L 1 (R d ),
along with

W 1 |ρ(t)| 2 π d2/2 , γ 2 π d2/2 C √ ln t , t e.
Theorem 1.7 can be informally restated by saying that

R d 1 |u(t, x ′ , x ′′ )| 2 dx ′
is dispersive in the variable x ′′ , with rate τ (t) -d2 . Such a dispersive property on a marginal was used in [START_REF] Antonelli | Scattering for nonlinear Schrödinger equation under partial harmonic confinement[END_REF] as a first step (following from Morawetz estimates) to prove scattering for NLS with a power-like nonlinearity and a partial confinement. In the present framework, no scattering result can be inferred.

Indeed, considering tensorized initial data of the form u 0 (x) = u 01 (x ′ )u 02 (x ′′ ), we have u(t, x) = u 1 (t, x ′ )u 2 (t, x ′′ ), where u 1 solves (1.1) with a fully confining potential, and u 2 solves (1.1) with V = 0, hence obeys Theorem 1.1 (with d replaced by d 2 ). In particular, u 1 may correspond to solutions described in Theorem 1.5, or be a Gaussian breather (see Section 3.3), showing that integrating with respect to x ′ first, in the above result, makes perfect sense, and the dynamics in the x ′ variable is independent of the dispersion stated in Theorem 1.7.

1.2.3. Repulsive harmonic potential. We finally present some results in the repulsive harmonic case,

(1.5) i∂ t u + 1 2 ∆u = -ω 2 |x| 2 2 u + λu ln |u| 2 , u |t=0 = u 0 , for x ∈ R d and ω, λ > 0.
In the case λ = 0, any defocusing energy-subcritical power-like nonlinearity |u| 2σ u, 0 < σ < 2 (d-2)+ , is short range for scattering, since the repulsive harmonic potential induces an exponential decay in time ( [START_REF] Carles | Nonlinear Schrödinger equations with repulsive harmonic potential and applications[END_REF]). In particular, in the dispersive frame (meaning after rescaling the wave function in terms of the time dependent dispersion, e ωt in this case), any asymptotic profile can be reached. On the other hand for ω = 0, Theorem 1.1 shows that there is only one profile (if one considers the modulus only) which can be reached in the dispersive frame. The case ω, λ > 0 is therefore a borderline case from these two perspectives: do we have scattering, or a universal behavior? In the next result, we give a partial answer to this question, showing the same dispersive rate as in the linear case λ = 0, and ruling out a universal behavior in the sense of Theorem 1.1. PROPOSITION 1.8. Let ω, λ > 0 and let τ -be the solution to the ODE

(1.6) τ-= ω 2 τ -+ 2λ τ - , τ -(0) = 1, τ-(0) = 0.
There exists

µ ∞ > 0 such that, as t → ∞, τ -(t) ∼ µ ∞ e ωt , τ-(t) ∼ ωµ ∞ e ωt . For u 0 ∈ Σ and the solution u ∈ L ∞ loc (R; Σ) to (1.5), consider the rescaling u(t, x) = 1 τ -(t) d/2 v t, x τ -(t) exp i τ-(t) τ -(t) |x| 2 2 .
Then v is bounded and not dispersive, in the sense that

sup t 0 R d 1 + |y| 2 + ln |v(t, y)| 2 |v(t, y)| 2 dy < ∞.
We can find two initial data

u 01 (x) = e -|x| 2 /2+iβ1|x| 2 /2 , u 02 (x) = e -|x| 2 /2+iβ2|x| 2 /2 , β j > 0,
such that the corresponding v j 's satisfy

|v j (t, •)| 2 -→ t→∞ γ 2 j in L 1 (R d ), with γ 1 = γ 2 .
In the same spirit as the comments following the statement of Theorem 1.7, we emphasize that the dynamics associated to the logarithmic nonlinearity (1.1) is completely different from the dynamics for NLS with a power-type nonlinearity. Typically, in [START_REF] Carles | Global existence results for nonlinear Schrödinger equations with quadratic potentials[END_REF], the case of a saddle potential, for x ∈ R d , d 2,

V (x) = ω 2 1 x 2 1 -ω 2 2 x 2 2 ,
was considered. There is confinement in the first direction, and a strong dispersion in the second one. In the case of a defocusing power-like nonlinearity |u| 2σ u, we have global existence in Σ. Given u 0 ∈ Σ, at least if σ 2/d and ω 2 is sufficiently large compared to ω 1 , there is scattering in the sense that there exist u ± ∈ Σ such that

U V (-t)u(t) -u ± Σ -→ t→±∞ 0, U V (t) = e -itH , H = - 1 2 ∆ + V.
Proposition 1.8 shows that for a logarithmic nonlinearity, there is still an exponential dispersion in x 2 , but the tensorization property implies that no scattering result can hold.

1.3. Outline of the paper. In Section 2, we show how to prove Proposition 1.3. In Section 3, we describe more precisely some remarkable properties related to the logarithmic nonlinearity in the presence of a quadratic potential: invariances, special transforms, and propagation of Gaussian data. Theorem 1.5 is proven in Section 4, relying on Lemma 4.2, which can be thought of as a dual to the logarithmic Sobolev inequality. In Section 5, we address the proof of Theorem 1.7, by showing essentially how to consider a suitable setting in order to relate this result to Theorem 1.1. Finally, Proposition 1.8 is proved in Section 6.

Cauchy problem

In this section, we briefly explain the proof of Proposition 1.3, which can be addressed like in the case without potential considered in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]. The main issue is that the logarithmic nonlinearity is not Lipschitz continuous at the origin. Proceeding as in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF], we first regularize the nonlinearity by saturating the logarithm near zero, and consider the sequence of approximate solutions given by (2.1)

i∂ t u ε + 1 2 ∆u ε = V (x)u ε + λ ln ε + |u ε | 2 u ε , u ε |t=0 = u 0 ,
where V , λ and u 0 are as in Proposition 1.3, and ε > 0. For fixed ε > 0, the above nonlinearity is locally Lipschitzian, with moderate growth at infinity, ensuring that it is L 2subcritical. Since the potential V is smooth and at most quadratic, local-in-time Strichartz estimates are available for H = -1 2 ∆+V , the same as in the case V = 0 (see e.g. [START_REF] Carles | Nonlinear Schrödinger equation with time dependent potential[END_REF] and references therein). Therefore, for every ε > 0, there is a unique, global solution at the L 2 level. Again, at fixed ε > 0, the nonlinearity is smooth, so higher regularity is propagated:

u ε , ∇u ε , xu ε ∈ C(R; L 2 (R d )).
The sequence (u ε ) 0<ε 1 converges, thanks to compactness arguments based on uniform a priori estimates. Since λ ∈ R and V is real-valued, the

L 2 -norm of u ε is indepen- dent of time, u ε (t) L 2 = u 0 L 2 . For 1 j d, differentiating (2.1) with respect to x j yields i∂ t ∂ j u ε + 1 2 ∆∂ j u ε = V (x)∂ j u ε + ∂ j V (x)u ε + λ ln ε + |u ε | 2 ∂ j u ε + 2λ 1 ε + |u ε | 2 Re(ū ε ∂ j u ε )u ε .
By assumption on V , |∂ j V (x)| 1 + |x|, so the standard L 2 estimate yields

d dt ∇u ε (t) 2 L 2 C u ε (t) 2 L 2 + xu ε (t) 2 L 2 + ∇u ε (t) 2 L 2 ,
where C is independent of ε. Similarly,

i∂ t (x j u ε ) + 1 2 ∆(x j u ε ) = ∂ j u ε + V (x)x j u ε + λ ln ε + |u ε | 2 x j u ε , hence d dt xu ε (t) 2 L 2 2 R d |xu ε (t, x)||∇u ε (t, x)|dx xu ε (t) 2 L 2 + ∇u ε (t) 2 L 2 .
In view of the conservation of the mass, Gronwall lemma implies that there exists C independent of ε such that

xu ε (t) 2 L 2 + ∇u ε (t) 2 L 2 C u 0 2 L 2 + xu 0 2 L 2 + ∇u 0 2
L 2 e C|t| , t ∈ R. Therefore, we have compactness in space for the sequence (u ε ) ε . Compactness in time follows from (2.1). Arzela-Ascoli theorem yields a converging subsequence, hence the existence part of Proposition 1.3.

The conservation of mass and energy can be proven like in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] (see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). Uniqueness follows from the remark that any solution

u ∈ L ∞ loc (R; Σ) to (1.1) actually belongs to C(R; L 2 (R d )), since the equation implies ∂ t u ∈ L ∞ loc (R; H -1 + F (H -1
)) (see [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]), and from the argument discovered in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF]: LEMMA 2.1 (Lemma 9.3.5 from [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). We have

Im z 2 ln |z 2 | 2 -z 1 ln |z 1 | 2 (z 2 -z1 ) 4|z 2 -z 1 | 2 , ∀z 1 , z 2 ∈ C . For two solutions u 1 , u 2 ∈ L ∞ loc (R; Σ) to (1.1), the difference w = u 2 -u 1 solves i∂ t w + 1 2 ∆w = V (x)w + λ ln |u 2 | 2 u 2 -ln |u 1 | 2 u 1 , w |t=0 = 0,
and the standard L 2 estimate yields, along with the above lemma, 1 2

d dt w(t) 2 L 2 (R d ) = λ Im R d ln |u 2 | 2 u 2 -ln |u 1 | 2 u 1 (ū 2 -ū1 )(t) dx 4λ w(t) 2 L 2 (R d ) , hence w ≡ 0 from Gronwall lemma.

Remarkable algebraic properties of (1.1)

3.1. Invariants. Apart from the conservations of mass and energy, (1.1) is invariant with respect to translation in time, but not in space in general, due to the potential V . When V is zero or exactly quadratic, there is a Galilean invariance (whose expression depends on the signature of ∇ 2 V ). Having the tensorization property in mind, recall the formulas for d = 1:

• Case V = 0: if u(t, x) solve (1.1), then for any v ∈ R, so does u(t, xvt)e ivx-iv 2 t/2 . • Case V (x) = ω 2 x 2 /2: if u(t, x) solve (1.1), then for any v ∈ R, so does u t, x -v sin(ωt) ω e ivx cos(ωt)-iv 2 cos(ωt) sin(ωt) 2ω .

• Case V (x) = -ω 2 x 2 /2: if u(t, x) solve (1.1), then for any v ∈ R, so does u t, x -v sinh(ωt) ω e ivx cosh(ωt)-iv 2 cosh(ωt) sinh(ωt) 2ω

.

The logarithmic nonlinearity causes a rather unique invariance, as noticed in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] for V = 0, a property which remains in the presence of V : If u solves (1.1), then for all k ∈ C, so does ku(t, x)e -itλ ln |k| 2 .

This shows that the size of the initial data alters the dynamics only through a purely time dependent oscillation, a feature which is fairly unusual for a nonlinear equation. In addition, considering k > 0, differentiating the above formula with respect to k and letting k → 0 shows that for t > 0 arbitrarily small, the flow map u 0 → u(t) cannot be C 1 , whichever function spaces are considered for u 0 and u(t), respectively; it is at most Lipschitzian.

Special potentials.

It is a standard fact that if V is linear in x, V (x) = E • x for some fixed E ∈ R d , the influence of V is explicit, in the sense that if u solves (1.1) with this V , and v solves (1.1) with V = 0 (and the same initial datum), then u and v are related through

v(t, x) = u t, x - t 2 2 E e i tE•x-t 3 3 |E| 2
.

See e.g. [START_REF] Carles | Nonlinear Schrödinger equation with time dependent potential[END_REF], where this formula is extended to the case where E depends on time.

On the other hand, lens transforms (see e.g. [START_REF] Carles | Nonlinear Schrödinger equation with time dependent potential[END_REF]) seem to be useless in the case of a logarithmic nonlinearity. If u solves (1.1) with V = 0, for ω > 0 and |t| < π/(2ω), set

(3.1) w + (t, x) = 1 (cos(ωt)) d/2 e -i ω 2 |x| 2 tan(ωt) u tan(ωt) ω , x cos(ωt)
.

Then w + solves

i∂ t w + + 1 2 ∆w + = λ cos(ωt) 2 ln cos(ωt) d |w + | 2 w + + ω 2 |x| 2 2 w + , w + |t=0 = u 0 .
The time dependent factor in front of the nonlinearity shows a strong difference with (1.1), and the equation in w + is not necessarily more pleasant to study. The case of a repulsive harmonic potential V (x) = -ω 2 |x| 2 /2 is similar (replace ω by iω).

Propagation of Gaussian initial data.

A remarkable feature of (1.1) is that when the potential is a polynomial of degree at most two in space, then an initial Gaussian data evolves as a Gaussian for all time. Plugging a time-dependent Gaussian function into the logarithmic nonlinearity, we readily see that this remarkable property is related to the well-known fact that the same holds in the case of the linear Schrödinger equation with potential; see e.g. [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF][START_REF] Hagedorn | Semiclassical quantum mechanics. III. The large order asymptotics and more general states[END_REF][START_REF] Heller | Time dependent approach to semiclassical dynamics[END_REF][START_REF] Hepp | The classical limit for quantum mechanical correlation functions[END_REF].

In view of the tensorization property described in the introduction, we consider the case d = 1, and a quadratic potential,

(3.2) i∂ t u + 1 2 ∂ 2 x u = λ ln |u| 2 u + Ω x 2 2 u, u |t=0 = u 0 ,
where Ω ∈ R is a constant. We seek u(t, x) = b(t)e -a(t)x 2 /2 (in particular u 0 is Gaussian). Plugging this into (3.2), we find:

i ḃ = 1 2 ab + λb ln |b| 2 ; i ȧ = a 2 + 2λ Re a -Ω.
Seeking a under the form

a = 1 τ 2 -i τ τ leads to (3.3) τ = 2λ τ + 1 τ 3 -Ωτ.
As long as the solution is smooth, multiply by τ and integrate in time:

(3.4) ( τ ) 2 = C 0 + 4λ ln τ - 1 τ 2 -Ωτ 2 .
In the case of a (constant) harmonic potential, we write Ω = ω 2 : τ is bounded, regardless of the sign of λ. Typically, we have generalized Gaussons for each sign of λ: u(t, x) = C ν e -kx 2 /2+iνt solves (3.2) if and only if

ω 2 -2λk = k 2 ; -ν - k 2 = λ ln(C 2 ν ).
For u to be an L 2 function in space, only one choice is possible for k,

k = -λ + λ 2 + ω 2 .
Like in [START_REF] Ferriere | The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superposition[END_REF], we see that τ is always periodic, and Gaussian data propagate as breathers in general, as a solitary wave in the specific case discussed above. Writing ω 2 = κ(κ + 2λ), κ > max(0, -2λ), we recover the generalized Gausson of the introduction, φ ν (x) = e -ν+κ/2 2λ e -κ|x| 2 /2 , with k = κ (and d = 1 here).

In the case of a (constant) repulsive harmonic potential, we write Ω = -ω 2 , and (3.3) becomes

(3.5) τ = ω 2 τ + 2λ τ + 1 τ 3 .
Assuming τ (0) > 0, we see that τ remains positive and bounded away from zero for all time, since (3.4) becomes

(3.6) τ (t) 2 = C 0 + ω 2 τ (t) 2 + 4λ ln τ (t) - 1 τ (t) 2 .
Therefore, in the case λ > 0, there exists δ > 0 such that

τ (t) δ, ∀t ∈ R. We infer trivially τ ω 2 δ, hence (3.7) τ (t) -→ t→∞ ∞, τ (t) -→ t→∞ ∞.
Therefore, we may approximate τ by the solution to

(3.8) τ = ω 2 τ.
In particular, the presence of the repulsive harmonic potential causes dispersion with an exponential dispersive rate. We analyze more precisely this situation in Section 6.

Full harmonic confinement

In this section, we prove Theorem 1.5, by following the strategy of Cazenave and Lions [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF], and proving some rigidity property of the generalized Gausson. We recall that

V (x) = κ(κ + 2λ) 2 |x| 2 , κ > 0,
and we denote ω 2 = κ(κ + 2λ) for conciseness.

4.1. Technical preliminary. We introduce the action and the Nehari functional:

S ν (u) := E(u) + ν u 2 L 2 , I ν (u) := ∇u 2 L 2 + ω 2 xu 2 L 2 + 2λ R d |u| 2 ln |u| 2 dx + 2ν u 2 L 2 = 2S ν (u) + 2λ u 2 L 2 .
We also define the quantity

D(ν) = inf{S ν (u) | u ∈ Σ(R d ) \ {0}, I ν (u) = 0} = -λ sup{ u 2 L 2 | u ∈ Σ(R d ) \ {0}, I ν (u) = 0}
, and the set of ground states by

G ν := {φ ∈ Σ(R d ) \ {0} | I ν (u) = 0, S ν (φ) = D(ν)}.
At this stage, we emphasize a major difference between the case of a power nonlinearity and the logarithmic nonlinearity. For power-type nonlinearities, it is standard to either minimize the action, or minimize the energy with a fixed mass, the two approaches being equivalent in the case of homogeneous nonlinearities. In the case of a logarithmic nonlinearity, the Nehari functional is constant along the flow of the equation, and minimizing the action without constraint, or the energy with a fixed mass, does not seem adequate. This characterization of the ground state in the case of a logarithmic nonlinearity is already present in [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF][START_REF] Ardila | Logarithmic Bose-Einstein condensates with harmonic potential[END_REF]. A key step of the analysis consists in showing that G ν = {e iθ φ ν , θ ∈ R}, with φ ν defined in Theorem 1.5,

φ ν (x) = e -ν+κd/2 2λ e -κ|x| 2 /2 .
First, we note that the energy functional E defined in (1.2) is of class C 1 , and for u ∈ Σ, its Fréchet derivative is given by

E ′ (u) = -∆u + ω 2 |x| 2 u + λu ln |u| 2 .
As a consequence, S ν and I ν are of class C 1 , and for u ∈ Σ S ′ ν (u), u = I ν (u). We will also need the following compactness result: LEMMA 4.1. For any sequence (u m ) uniformly bounded in Σ, there exists a subsequence (still denoted (u m )) and some u ∈ Σ such that:

• u m -→ m→∞ u in L 2 (R d ),
• u m -→ m→∞ u a.e. in R d , • The following convergence also holds:

lim m→∞ R d |u m | 2 ln |u m | 2 dx = R d |u| 2 ln |u| 2 dx.
SKETCH OF THE PROOF. The first two points are standard. For the third point, using the fact that for all δ > 0, there exists C δ > 0 such that for all y 0

|y 2 ln y 2 | C δ (y 2-δ + y 2+δ ), we get, R d |u m -u| 2 ln |u m -u| 2 dx C 1 + u m 2 Σ + u 2 Σ u m -u L 2 -→ m→∞ 0.
The third point then follows from Brézis-Lieb lemma [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF], in the convex case (see also [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF]Lemma 2.3]).

Variational analysis.

The main novelty to prove orbital stability in the case λ > 0 is a result which may be viewed as a dual of the celebrated logarithmic Sobolev inequality: LEMMA 4.2. Let f ∈ F (H 1 (R d )) and a > 0:

(4.1) - R d |f (x)| 2 ln |f (x)| 2 f 2 L 2 dx a R d |x| 2 |f (x)| 2 dx + d 2 f 2 L 2 ln π a .

There is equality if and only if |f

(x)| = ce -a|x| 2 /2 , with c = f L 2 (a/π) d/2 . Assuming f L 2 = 1, we also have - R d |f (x)| 2 ln |f (x)| 2 dx d 2 ln 2eπ d R d |x| 2 |f (x)| 2 dx .
REMARK 4.3. The above inequalities are not translation invariant, just like the usual logarithmic Sobolev inequality is not translation invariant on the Fourier side. However, we must take much more than translation on the Fourier side into account, as the inequality is obviously invariant by multiplication by e iφ(x) for φ real-valued, not necessarily linear in x.

The second inequality is the counterpart of the classical optimal logarithmic Sobolev inequality, for

f L 2 = 1, R d |f (x)| 2 ln |f (x)| 2 dx d 2 ln d 2eπ ∇f 2 L 2 ,
established in [START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF].

PROOF. For a > 0, consider the normalized Gaussian

ν a (x) = a π d/2 e -a|x| 2 .
It is normalized to ensure that ν a is a probability density. In view of the Csiszár-Kullback inequality (see e.g. [1, Th. 8.2.7]), for µ, ν probability densities,

µ -ν 2 L 1 (R d ) 2 R d µ(x) ln µ(x) ν(x) dx. Let µ(x) = |f (x)| 2 / f 2 L 2
and ν = ν a : since the above relative entropy (right-hand side) is non-negative, we have

R d |f (x)| 2 ln |f (x)| 2 f 2 L 2 ν a (x) dx 0.
Replacing ν a by its explicit value and expanding yields (4.1).

Moreover, equality holds in (4.1) if and only if the relative entropy is zero, and Csiszár-Kullback inequality implies that µ(x) = ν a (x).

Suppose f L 2 = 1. The map a → αa -β ln a + c, α, β > 0 reaches its minimum for a = β/α, so we optimize (4.1)

with α = xf 2 L 2 , β = d 2 and c = d 2 ln π, - R d |f (x)| 2 ln |f (x)| 2 dx d 2 ln 2e d xf 2 L 2 + d 2 ln π,
hence the announced inequality.

LEMMA 4.4 (Ground energy and ground states). Let ν ∈ R. Then the quantity D(ν) is given by

D(ν) = -λ φ ν 2 L 2 = -λπ d 2 κ -d 2 e -ν+κd/2 λ .
Moreover,

G ν = e iθ φ ν , θ ∈ [0, 2π[ . PROOF. Let u ∈ Σ \ {0} such that I ν (u) = 0.
First, we apply Lemma 4.2 with a = κ:

(4.2) 2λκ xu 2 L 2 + 2λ R d |u(x)| 2 ln |u(x)| 2 dx 2λ u 2 L 2 ln u 2 L 2 - d 2 ln π κ .
Using the fact that ω 2 = κ 2 + 2λκ, we thus get

(4.3) 0 = I ν (u) ∇u 2 L 2 + κ 2 xu 2 L 2 + 2λ u 2 L 2 ln u 2 L 2 - d 2 ln π κ + 2ν u 2 L 2 .
Moreover, since the ground state of the harmonic oscillator -∆ + κ 2 |x| 2 is e -κ|x| 2 /2 , the bottom of the spectrum is κd, thus (4.4)

∇u 2 L 2 + κ 2 xu 2 L 2
κd u 2 L 2 , with equality if and only if there exists µ ∈ C such that u(x) = µ e -κ|x| 2 /2 . Thus, there holds

0 2λ u 2 L 2 ln u 2 L 2 - d 2 ln π κ + κd 2λ + ν λ .
Therefore, since u = 0, we get:

ln u 2 L 2 - d 2 ln π κ + κd 2λ + ν λ 0, which yields (4.5) u 2 L 2 π d 2 κ -d 2 e -ν λ -κd 2λ .
Since this inequality holds for all u ∈ Σ \ {0} such that I ν (u) = 0, we get

D(ν) -λπ d 2 κ -d 2 e -ν λ -κd 2λ .
Moreover, there is equality in this last inequality if and only if there is equality in all the previous inequalities (4.2), (4.4) and (4.5). From the previous discussion, equality in both (4.2) and (4.4) is equivalent to u(x) = µ e -κ|x| 2 /2 for some µ ∈ C. Putting this condition in the last case of equality shows that |µ| 2 = e -ν+κd/2 λ . Therefore, all the inequalities are equalities if and only if there exists θ ∈ [0, 2π[ such that u = e iθ φ ν . The conclusion then readily follows. REMARK 4.5. If we had to consider equality in (4.2) only, then θ would be a map from R d to R, not necessarily constant. On the other hand, equality in (4.4) forces θ to be constant.

Minimizing sequences.

As a final preparation for the proof of Theorem 1.5, we characterize minimizing sequences. LEMMA 4.6. {u ∈ Σ \ {0}, I ν (u) = 0} is a bounded subset of Σ.

PROOF. Let u ∈ Σ \ {0} such that I ν (u) = 0. By definition of D(ν), we know that

u 2 L 2 -λ -1 D(ν).
Moreover, using (4.3), we get

∇u 2 L 2 + κ 2 xu 2 L 2 -2λ u 2 L 2 ln u 2 L 2 + 2λ u 2 L 2 d 2 ln π κ + 2ν .
Using the fact that -x ln x e -1 for all x 0, we get

∇u 2 L 2 + κ 2 xu 2 L 2 2λe -1 -2D(ν) d 2 ln π κ + 2ν ,
hence the conclusion, since the L 2 -norm is controlled by -D(ν).

LEMMA 4.7. Let (u n ) be a minimizing sequence for D(ν). Then there exists φ ∈ G ν such that, up to a subsequence,

u n -φ Σ -→ n→∞ 0.

PROOF. By definition of D(ν), we know that

u n 2 L 2 -→ n→∞ -λ -1 D(ν).
Moreover, Lemmas 4.6 and 4.1 show that there exists φ ∈ Σ such that, up to a subsequence,

u n ⇀ φ
weakly in Σ,

u n → φ strongly in L 2 (R d ), lim n→∞ R d |u n | 2 ln |u n | 2 dx = R d |φ| 2 ln |φ| 2 dx.
In particular, this yields

φ L 2 = -λ -1 D(ν).
Moreover, using again (4.3), there holds

∇u n 2 L 2 + κ 2 xu n 2 L 2 -2λ u n 2 L 2 ln u n 2 L 2 + 2λ u n 2 L 2 d 2 ln π κ - ν λ ,
and

-2λ u n 2 L 2 ln u n 2 L 2 + 2λ u n 2 L 2 d 2 ln π κ - ν λ -→ n→∞ 2D(ν) ln -λ -1 D(ν) - d 2 ln π κ + ν λ = - κd λ D(ν),
by using Lemma 4.4. Thus, using again (4.4), we get

- κd λ D(ν) = κd φ L 2 ∇φ 2 L 2 + κ 2 xφ 2 L 2 lim inf ∇u n 2 L 2 + κ 2 xu n 2 L 2 lim sup ∇u n 2 L 2 + κ 2 xu n 2 L 2 - κd λ D(ν).
Therefore, there is equality in all of the above inequalities. In particular,

∇φ 2 L 2 + κ 2 xφ 2 L 2 = lim ∇u n 2 L 2 + κ 2 xu n 2 L 2 , which proves that u n → φ strongly in Σ.
With all the previous convergences, the convergence of I ν (u n ) to I ν (φ) is now obvious, and therefore

I ν (φ) = 0. Since D(ν) = -λ φ L 2 , we obtain φ ∈ G ν .
4.4. Orbital stability. Theorem 1.5 is then classically proved by contradiction. Assume that there exist ε > 0, u 0n ∈ Σ and t n ∈ R such that

inf θ∈R u 0n -e iθ φ ν Σ -→ n→∞ 0, (4.6) inf θ∈R u n (t n ) -e iθ φ ν Σ > ε for any n ∈ N. (4.7) Set v n = u n (t n )
. By (4.6) and the conservation laws, we obtain

v n 2 L 2 = u 0n 2 L 2 -→ n→∞ φ ν 2 L 2 , E(v n ) = E(u 0n ) -→ n→∞ E(φ ν ).
In particular, there also holds

S ν (v n ) -→ n→∞ S ν (φ ν ) = D(ν). Since I ν (f ) = 2E(f ) + 2(λ + ν) u 2 L 2 , I ν (v n ) -→ n→∞ I ν (φ ν ) = 0.
Next, define the sequence f n = ρ n v n with

ρ n = exp - I ν (v n ) 2 v n 2 L 2
.

It is clear that I ν (f n ) = 0 and lim n ρ n = 1, so that v n -f n Σ → 0 and thus S ν (f n ) → D(ν). Therefore, (f n ) is a minimizing sequence for D(ν). Applying Lemma 4.7, there exists φ ∈ G ν such that, up to a subsequence, f n → φ strongly in Σ. Therefore (4.8)

v n → φ strongly in Σ.
Moreover, since φ ∈ G ν , Lemma 4.4 gives the existence of θ ∈ [0, 2π[ such that φ = e iθ φ ν . Therefore, (4.8) contradicts (4.7).

Partial harmonic confinement

5.1. Preparation of the proof. In this section, we explain how to prove Theorem 1.7, by adapting elements of proof introduced in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF][START_REF] Ferriere | Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation[END_REF]. Like in Theorem 1.1, we rescale the initial unknown u to a new unknown v through the formula

(5.1) u(t, x ′ , x ′′ ) = 1 τ (t) d2/2 v t, x ′ x ′′ τ (t) u 0 L 2 (R d ) γ L 2 (R d 2 ) exp i τ (t) τ (t) |x ′′ | 2 2 ,
where we emphasize that at this stage, γ is a function of the d 2 -dimensional variable x ′′ . The function ρ involved in Theorem 1.7 is then simply

ρ(t, y) = R d 1 |v(t, x ′ , y)| 2 dx ′ , y ∈ R d2 .
We readily check that (1.1) is then equivalent to

i∂ t v+ 1 2 ∆ x ′ v+ 1 2τ (t) 2 ∆ y v = λ ln |v| 2 v+ ω 2 2 |x ′ | 2 + λ|y| 2 v+θ(t)v, v |t=0 = αu 0 , with α = γ L 2 (R d 2 ) / u 0 L 2 (R d )
, and where θ is real-valued and depends on time only,

θ(t) = -λ d 2 ln τ (t) -2 ln u 0 L 2 (R d ) γ L 2 (R d 2 ) = -λ d 2 ln τ (t) -2 ln u 0 L 2 (R d ) π d2/2 .
Changing v(t, x ′ , y) to v(t, x ′ , y)e i t 0 θ(s)ds does not affect the function ρ involved in the statement of Theorem 1.7, and removes the last term from the above equation, so we may assume that v solves

(5.2) i∂ t v + 1 2 ∆ x ′ v + 1 2τ (t) 2 ∆ y v = λ ln |v| 2 v + ω 2 2 |x ′ | 2 + λ|y| 2 v, v |t=0 = v 0 , where v 0 is such that v 0 L 2 (R d ) = γ L 2 (R d 2 )
. The equation (5.2) is still Hamiltonian, but since it is no longer autonomous, we have a dissipated energy: setting

E(t) = 1 2 ∇ x ′ v 2 L 2 (R d ) + 1 2τ (t) 2 ∇ y v 2 L 2 (R d ) + λ R d |v(t, x ′ , y)| 2 ln |v(t, x ′ , y)| 2 dx ′ dy + R d ω 2 2 |x ′ | 2 + λ|y| 2 |v(t, x ′ , y)| 2 dx ′ dy, we have dE dt = - τ τ 3 ∇ y v 2 L 2 (R d ) 0.
This implies crucial a priori estimates:

LEMMA 5.1. Let u 0 ∈ Σ, λ, ω > 0. There exists C > 0 such that

sup t 0 R d 1 + |x ′ | 2 + |y| 2 + ln |v(t, x ′ , y)| 2 |v(t, x ′ , y)| 2 dx ′ dy C, sup t 0 1 2 ∇ x ′ v 2 L 2 (R d ) + 1 2τ (t) 2 ∇ y v 2 L 2 (R d ) C. In addition, ∞ 0 τ (t) τ (t) 3 ∇ y v(t) 2 L 2 (R d ) < ∞.
SKETCH OF THE PROOF. We resume the main lines of the proof of [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]Lemma 4.1]. Write E = E + -E -, where

E + = 1 2 ∇ x ′ v 2 L 2 + 1 2τ (t) 2 ∇ y v 2 L 2 + R d ω 2 2 |x ′ | 2 + λ|y| 2 |v(t, x ′ , y)| 2 dx ′ dy + λ |v(t,x ′ ,y)| 2 1 |v(t, x ′ , y)| 2 ln |v(t, x ′ , y)| 2 dx ′ dy.
Note that E + is the sum of non-negative terms only. Since E is non-increasing, for t 0,

E + (t) E(0) + E -(t) = E(0) + λ |v| 1 |v(t, x ′ , y)| 2 ln 1 |v(t, x ′ , y)| 2 dx ′ dy 1 + R d |v(t, x ′ , y)| 2-ε dx ′ dy, for any ε > 0 sufficiently small. Then v L 2-ε v 1-δ(ε) L 2 (|x ′ | + |y|)v δ(ε)
L 2 with δ(ε) → 0 as ε → 0. In view of the conservation of the L 2 -norm of v (which is equal to the L 2 -norm of u), we infer, for 0

< ε ≪ 1 , E -E 1/2 + (for instance), hence E + (t) 1, ∀t 0.
Therefore, each term in E + is bounded, and so is E -.

Since E is bounded, the integral of Ė is bounded, hence the last part of the lemma.

Convergence of momenta.

The proof of (1.7) stems from the same arguments as in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF].

5.2.1. Center of mass. Introduce

I 1 (t) := Im R d v(t, x ′ , y) ∇ y v(t, x ′ , y)dx ′ dy, I 2 (t) := R d y |v(t, x ′ , y)| 2 dx ′ dy.
We compute

İ1 = -2λI 2 , İ2 = 1 τ 2 (t) I 1 ,
and so Ĩ2 := τ I 2 satisfies Ï2 = 0. Since τ (t) ∼ 2t √ λ ln t as t → ∞, we infer that I 2 (t) = O(ln t) -1/2 . 5.2.2. Second order momentum. Rewriting in terms of v the conservation of the energy for u, (1.2), we find:

E(u 0 ) = 1 2 ∇ x ′ v 2 L 2 (R d ) + 1 2τ (t) 2 ∇ y v 2 L 2 (R d ) + ( τ ) 2 2 R d |y| 2 |v(t, x ′ , y)| 2 dx ′ dy + τ τ Im R d v(t, x ′ , y)y • ∇ y v(t, x ′ , y)dx ′ dy + ω 2 2 R d |x ′ | 2 |v(t, x ′ , y)| 2 dx ′ dy + λ R d |v(t, x ′ , y)| 2 ln |v(t, x ′ , y)| 2 dx ′ dy -λd 2 π d2 ln τ + 2λ u 0 2 L 2 (R d ) ln u 0 L 2 (R d ) π d2/2 .
In view of Lemma 5.1, the first, second, fifth, sixth, and last terms on the right-hand side are bounded in time. In view of Lemma 5.1 and Cauchy-Schwarz inequality, the fourth term is O( τ ) = O( √ ln t), and we obtain

( τ ) 2 2 R d |y| 2 |v(t, x ′ , y)| 2 dx ′ dy -λd 2 π d2 ln τ = O( √ ln t),
which yields, since ( τ ) 2 = 4λ ln τ (multiply (1.3) by τ and integrate),

R d |y| 2 |v(t, x ′ , y)| 2 dx ′ dy = d 2 2 π d2 + O 1 √ ln t = R d 2 |y| 2 γ(y) 2 dy + O 1 √ ln t ,
hence (1.7).

Convergence of the profile.

To prove the end of Theorem 1.7, we use a Madelung transform: define R, J 1 and J 2 by

R(t, x ′ , y) = |v(t, x ′ , y)| 2 , J 1 (t, x ′ , y) = Im (v(t, x ′ , y)∇ x ′ v(t, x ′ , y)) , J 2 (t, x ′ , y) = Im (v(t, x ′ , y)∇ y v(t, x ′ , y)) .
We check that (5.2) implies

∂ t R + ∇ x ′ • J 1 + 1 τ (t) 2 ∇ y • J 2 = 0, ∂ t J 2 + λ∇ y R + 2λyR = 1 4τ (t) 2 ∆ y ∇ y R - 1 τ (t) 2 ∇ y • (Re (∇ y v ⊗ ∇ y v)) + 1 2 ∇ x ′ • Re (v ((∇ x ′ ⊗ ∇ y ) v) -∇ x ′ v ⊗ ∇ y v) .
We do not write the evolution law for J 1 , as it is not needed for the argument.

REMARK 5.2. Despite the fact that v(t) might not be H 2 , the term v (∇ x ′ ⊗ ∇ y ) v is still well defined in D ′ ((0, ∞) × R d ) owing to the fact that v is in L ∞ loc ((0, ∞); H 1 ) and to the relation:

v (∇ x ′ ⊗ ∇ y ) v = ∇ y (v∇ x ′ v) -∇ y v ⊗ ∇ x ′ v.
We note that

ρ(t, y) = R d 1 R(t, x ′ , y)dx ′ ,
and we set

j(t, y) = R d 1 J 2 (t, x ′ , y)dx ′ . These new unknowns solve, in D ′ ((0, ∞) × R d2 ):        ∂ t ρ + 1 τ (t) 2 ∇ y • j = 0, ∂ t j + λ∇ y ρ + 2λyρ = 1 4τ (t) 2 ∆ y ∇ y ρ - 1 τ (t) 2 ∇ y • µ, where µ(t, y) = R d 1 Re (∇ y v ⊗ ∇ y v) (t, x ′ , y)dx ′ .
At this stage, in view of the a priori estimates provided by Lemma 5.1, we have the same ingredients are those needed in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]Section 5.3] and [START_REF] Ferriere | Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation[END_REF]Lemma 4.4], hence the weak convergence and the estimate of the Wasserstein distance in Theorem 1.7.

Repulsive harmonic potential

In this section, we consider (1.5), that is

i∂ t u + 1 2 ∆u = -ω 2 |x| 2 2 u + λu ln |u| 2 , u |t=0 = u 0 ,
for x ∈ R d and ω, λ > 0.

6.1. Analysis of the dispersion. We first resume the analysis started in Section 3.3 in the Gaussian case, and consider τ solving (3.5). We discuss the dependence upon initial data at the end. In view of (3.7), for t T ≫ 1,

|e T (t)| ε t T sinh (ω(t -s)) ω ds ε ω 2 cosh (ω(t -T )) .
On the other hand,

τ T eff (t) ∼ t→∞ 1 2 τ (T ) + τ (T ) ω e ω(t-T ) .
We infer from (3.7) that τ grows exponentially in time, since τ (T ) + τ (T ) ω ≫ ε. In view of (3.6), τ also grows exponentially in time: (6.1) e ωt τ (t) e ωt , e ωt τ (t) e ωt , t > 0. The map t → μ(t)e 2ωt is increasing, and

μ(t)e 2ωt t -c -→ t→∞ ∞. Then (6.2) implies μ(t)e 2ωt -t + c -→ t→∞ -∞.
Therefore, for t sufficiently large, μ is positive decreasing, hence has a non-negative limit as t → ∞, and μ ∈ L 1 . But (6.2) implies that μ and μ are simultaneously L 1 , so μ ∈ L 1 and the limit of μ has to be zero. This yields the existence of µ ∞ > 0 such that

µ(t) -→ t→∞ µ ∞ , hence τ (t) ∼ t→∞ µ ∞ e ωt .
In view of (3.6), and since τ > 0 (at least for t ≫ 1), we infer We also already know that τ (t) ∼ µ ∞ e ωt , so in particular F (t) ∼ 2λ µ∞ e -ωt , and thus Therefore, (6.4) leads to an expression for µ ∞ :

τ (t) ∼ t→∞ ωµ ∞ e ωt .
(6.5) µ ∞ = τ (0) + τ (0) ω + 1 2ω ∞ 0 e -ωr 2λ τ (r) + 1 τ (r) 3 dr.
Moreover, from the fact that F 0 and e ωt e -ωr -e -ωt e ωr 0 for all 0 r t, (6.4) also gives for all t 0:

τ (t) τ (0) cosh(ωt) + τ (0) ω sinh(ωt).
Assume τ (0) = τ 1 0 and τ (0) = τ 0 > 0. Then, the last term in the right-hand side of (6.5) can be estimated. First:

0 ∞ 0 e -ωr 2λ τ (r) dr ∞ 0 e -ωr 2λ τ 0 cosh(ωr) + τ1 ω sinh(ωr) dr ∞ 0 e -2ωr 4λ τ 0 (1 + e -2ωr ) + τ1 ω (1 -e -2ωr ) dr - 2λ ω(τ 0 -τ1 ω ) ln τ 0 (1 + e -2ωr ) + τ 1 ω (1 -e -2ωr ) ∞ 0 2λ ω( τ1 ω -τ 0 ) ln 1 + τ1 ω -τ 0 2τ 0 .
Then, we also have

0 ∞ 0 e -ωr 1 (τ (r)) 3 dr ∞ 0 e -ωr 1
τ 0 cosh(ωr) + τ1 ω sinh(ωr)

3 dr ∞ 0 e 2ωr 16λ
τ 0 (1 + e 2ωr ) + τ1 ω (e 2ωr -1)

3 dr - 4λ ω(τ 0 + τ1 ω ) τ 0 (1 + e 2ωr ) + τ 1 ω (e 2ωr -1) -2 ∞ 0 2λ ω( τ1 ω + τ 0 ) τ 0 .
Then, as soon as τ 0 is fixed, we get an expansion of µ ∞ with respect to τ 1 with (6.5):

(6.6) µ ∞ = τ 1 ω + τ 0 + O ln τ 1 τ 1 .
6.2. Back to the PDE. We now address the general case, in the sense that u 0 need not be Gaussian. As announced in Proposition 1.8, change the unknown u to v, through the formula

(6.7) u(t, x) = 1 τ -(t) d/2 v t, x τ -(t) exp i τ-(t) τ -(t) |x| 2 2 ,
with τ -solution to (1.6), that is, (3.5) where the last term is discarded (this simplifies a little bit the computations, and the discarded term brings no extra information any way). Then (3.2) is equivalent, up to an irrelevant time dependent phase (like previously), to (6.8)

i∂ t v + 1 2τ 2 - ∆v = λ|y| 2 v + λv ln |v| 2 , v |t=0 = u 0 ,
provided that we assume (6.9)

τ -(0) = 1, τ-(0) = 0. L 2 with δ(ε) → 0 as ε → 0, and we conclude like in the proof of Lemma 5.1.

Since the L 2 -norm of v is independent of time, the boundedness of the momentum of v is an evidence that v is not dispersive, while the boundedness of |v| 2 in LlogL is an evidence that v does not grow to infinity. Therefore τ -describes the dispersive rate of any solution to (1.5).

6.2.2. Center of mass. Introduce Set Ĩ2 := τ -I 2 . We compute, in view of (3.5),

d 2 dt 2 Ĩ2 = ω 2
Ĩ2 , hence τ -(t)I 2 (t) = a 0 cosh(ωt) + b 0 sinh(ωt) ω .

In view of (6.9), a 0 = I 2 (0), b 0 = İ2 (0) = I 1 (0).

We infer

I 2 (t) ∼ t→∞ a 0 + b 0 2 e ωt τ -(t) ∼ t→∞ a 0 + b 0 2µ ∞ = I 1 (0) + I 2 (0) 2µ ∞ .
Unlike in the case ω = 0 considered in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF][START_REF] Ferriere | Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation[END_REF], the asymptotic center of mass of v is not zero in general (while it is always zero in the context of Theorem 1.1); this is like in the scattering case, where the asymptotic profile has no reason to be centered at the origin (as shown by the existence of wave operators).

Integrating in time the first equation in (6.10), we infer

I 1 (t) ∼ t→∞ -λt I 1 (0) + I 2 (0) µ ∞ ,
provided that I 1 (0) + I 2 (0) = 0. This suggests that v is oscillatory.

6.2.3. More on large time behavior. At this stage, we have established two differences with the dynamics of logNLS (without potential): the dispersion is the one dictated by the repulsive harmonic potential, that is, exponential, and in the dispersive frame (working with v), the asymptotic center of mass is not necessarily zero.

The next natural question would be to decide between a general asymptotic behavior for |v| (like in the case ω = 0) or a complete scattering theory. We are not able to fully validate the second option, which is the most likely in view of the result on the center of mass, but the Gaussian case shows that no universal profile must be expected for the large time behavior of |v|.

To see this, consider d = 1 and two Gaussian initial data u 01 (x) = e -x 2 /2+iβ1x 2 /2 , u 02 (x) = e -x 2 /2+iβ2x 2 /2 , β j > 0.

In other words, we start from two distinct Gaussian data, whose moduli (hence all Lebesgue norms and momenta, for instance) are equal. The corresponding solutions u 1 and u 2 are given by the formula presented in Section 3.3, boiling the description down to the analysis of the ODE (3.5):

u j (t, x) = b j (t)e -x 2 /(2τj (t) 2 )+i τj (t)x 2 /(2τj (t)) , τ j (t) ∼ t→∞ µ ∞,j e ωt .

In view of Section 6.1, we have

µ ∞,j = 1 + β j ω + O ln β j β j as β j → ∞,
and so for β 2 ≫ β 1 ≫ 1, the corresponding functions v j have different (asymptotic centers of mass and) asymptotic profiles. The fact that |v j | 2 converges strongly in L 1 to the corresponding limiting Gaussian is straightforward; see e.g. [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]Corollary 1.11].

6. 1 . 1 .

 11 Direct error estimate. Let τ T eff solve τ T eff = ω 2 τ T eff , τ T eff (T ) = τ (T ), τ T eff (T ) = τ (T ). Consider the error e T = τ -τ T eff . It solves ëT -ω 2 e T

6. 1 . 3 . 0 F 0 e -2ωs s 0 F 0 F

 130000 Dependence of µ ∞ . First, we get another integral expression for τ . Define F (t) := 2λ τ (t) + 1 τ (t)3 . Then there holdsτ -ω 2 τ = F, which leads to d dt e ωt ( τ -ωτ ) = F (t) e ωt ,and thuse ωt ( τ -ωτ ) = ( τ (0) -ωτ (0)) + t (s)e ωs ds.Therefore, we getτ (t) = e ωt τ (0) + 1 -e -2ωt 2ω ( τ (0) -ωτ (0)) + t (r)e ωr drds .After some easier computations, we get (6.4) τ (t) = τ (0) cosh(ωt) + τ (r)e ωr dr.

t 0 F 0 F

 00 (r)e ωr dr = O(t), ∞ (r)e -ωr dr < ∞.

I 1 (

 1 t) := Im R d v(t, y)∇v(t, y)dy , I 2 (t) := R d y|v(t, y)| 2 dy .

  |y| 2 |v(t, y)| 2 dy + λ R d |v(t, y)| 2 ln |v(t, y)| 2 dy.We readily infer, with the same proof as in the case ω = 0 given in[START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] Lemma 4.1], as already sketched in the proof of Lemma 5.1: LEMMA 6.1. For u 0 ∈ Σ and λ > 0, there holds SKETCH OF THE PROOF. Write E = E + -E -, whereE + = 1 2τ -(t) 2 ∇v(t) 2 L 2 + λ R d |y| 2 |v(t, y)| 2 dy + λ |v|>1 |v(t, y)| 2 ln |v(t, y)| 2 dy.Note that E + is the sum of non-negative terms only. Since E is non-increasing, for t 0,E + (t) E(0) + E -(t) = E(0) + λ |v(t, y)| 2-ε dy,for any ε > 0 sufficiently small. Then v L 2-ε v

	6.2.1. Hamiltonian structure and consequences. Set E(t) = 1 2τ -(t) 2 ∇v(t) 2 L 2 =:E kin (t) +λ R d We compute Ė(t) = -2 τ-(t) τ -(t) E kin (t). sup t 0 R d 1 + |y| 2 + ln |v(t, y)| 2 |v(t, y)| 2 dy + τ -(t) 2 ∇v(t) 2 1 L 2 (R d ) and ∞ 0 τ-(t) τ 3 -(t) ∇v(t) 2 |v(t, y)| 2 ln 1 |v(t, y)| 2 dy 1 + R d L |v| 1 1-δ(ε) L 2 yv δ(ε)	< ∞,

2 (R d ) dt < ∞.