A semilinear problem associated to the space-time fractional heat equation in R<i><SUP>N</SUP></i>
| dc.contributor.author | Cortazar, Carmen | |
| dc.contributor.author | Quiros, Fernando | |
| dc.contributor.author | Wolanski, Noemi | |
| dc.date.accessioned | 2025-01-20T16:05:45Z | |
| dc.date.available | 2025-01-20T16:05:45Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | We study the fully nonlocal semilinear equation partial derivative(alpha )(t)u +(-Delta)(beta )u = |u|(p-1 )u, p >= 1, where partial derivative t alpha stands for the usual time derivative when alpha=1 and for the Caputo alpha-derivative if alpha is an element of (0, 1), while (-Delta)(beta), beta is an element of (0, 1], is the usual beta power of the Laplacian. We prescribe an initial datum in L-q(R-N). We give conditions ensuring the existence and uniqueness of a solution living in L-q(R-N) up to a maximal existence time T that may be finite or infinite. If T is finite, the L-q norm of the solution becomes unbounded as time approaches T, and u is said to blow up in L-q. Otherwise, the solution is global in time. For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. Our weakest condition for global existence and our condition for blow-up are both related to the size of the averages of the initial datum in balls. As a corollary, every nonnegative nontrivial solution in L-q blows up in finite time if 1 < p < p(f ):= 1 + 2 beta/N whereas if p > p(f) there are both solutions that blow up and global ones. Noteworthy, the critical Fujita-type exponent pf does not depend on alpha. However, there is an important difference in the behavior of solutions in the critical case p = p(f) depending on the value of this parameter: when alpha = 1 it was known that all nonnegative and nontrivial solutions blow up, while we prove here that if alpha is an element of (0, 1) there is global existence for some initial data. | |
| dc.fuente.origen | WOS | |
| dc.identifier.doi | 10.1007/s00526-024-02836-z | |
| dc.identifier.eissn | 1432-0835 | |
| dc.identifier.issn | 0944-2669 | |
| dc.identifier.uri | https://doi.org/10.1007/s00526-024-02836-z | |
| dc.identifier.uri | https://repositorio.uc.cl/handle/11534/89923 | |
| dc.identifier.wosid | WOS:001336854500002 | |
| dc.issue.numero | 9 | |
| dc.language.iso | en | |
| dc.revista | Calculus of variations and partial differential equations | |
| dc.rights | acceso restringido | |
| dc.title | A semilinear problem associated to the space-time fractional heat equation in R<i><SUP>N</SUP></i> | |
| dc.type | artículo | |
| dc.volumen | 63 | |
| sipa.index | WOS | |
| sipa.trazabilidad | WOS;2025-01-12 |