$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups

Santiago Gómez Cobos; Joel E. Restrepo; Michael Ruzhansky

doi:10.5802/crmath.643

Research article - Partial differential equations

L^{p} - L^{q}

estimates for non-local heat and wave type equations on locally compact groups

Santiago Gómez Cobos ¹ ; Joel E. Restrepo ^{1
,} ² ; Michael Ruzhansky ^{1
,} ³

¹ Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, B 9000 Ghent, Belgium
² Department of Mathematics, Cinvestav IPN, Mexico city, Mexico
³ School of Mathematical Sciences, Queen Mary University of London, United Kingdom

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1331-1336

Abstract (Original language)
Résumé

We prove the $L^{p} - L^{q}$ $(1 < p ⩽ 2 ⩽ q < + \infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any positive left invariant operator (maybe unbounded) on $G$ . We complement our studies by giving asymptotic time estimates for the solutions, which in some cases are sharp.

On montre les estimations de norme $L^{p} - L^{q}$ $(1 < p ⩽ 2 ⩽ q < + \infty)$ pour les solutions des équations dites « de type chaleur » et « de type onde » définies sur un groupe localement compact, séparable et unimodulaire $G$ en utilisant un opérateur intégro-différentiel sur le temps et un opérateur positif invariant á gauche quelconque sur $G .$ De plus, on donne des estimations de temps asymptotiques pour ces solutions, qui deviennent des estimations optimales dans quelques cas.

Received: 2023-06-22
Accepted: 2024-04-21
Published online: 2024-11-14

Zbl

DOI: 10.5802/crmath.643

Classification: 43A15, 43A85, 45K05
Keywords: Locally compact groups, heat type equations, wave type equations, asymptotic estimates, non-local operators
Mots-clés : Groupes localement compacts, équations de type chaleur, équations de type onde, estimations asymptotiques, opérateurs non locaux

Author's affiliations:

Santiago Gómez Cobos ¹ ; Joel E. Restrepo ^{1
,

2} ; Michael Ruzhansky ^{1
,

3}

¹ Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, B 9000 Ghent, Belgium
² Department of Mathematics, Cinvestav IPN, Mexico city, Mexico
³ School of Mathematical Sciences, Queen Mary University of London, United Kingdom

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2024__362_G11_1331_0,
     author = {Santiago G\'omez Cobos and Joel E. Restrepo and Michael Ruzhansky},
     title = {$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1331--1336},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     doi = {10.5802/crmath.643},
     zbl = {07945476},
     language = {en},
}

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Santiago Gómez Cobos; Joel E. Restrepo; Michael Ruzhansky. $L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1331-1336. doi: 10.5802/crmath.643

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