Decay estimates for nonlocal problems via energy methods
By Liviu I. Ignat, Julio D. Rossi. 2009
63 citations recorded by the source corpus.
Abstract In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form u t ( x , t ) = ∫ R d G ( x − y ) ( u ( y , t ) − u ( x , t ) ) d y . For example, we will consider equations like, u t ( x , t ) = ∫ R d J ( x , y ) ( u ( y , t ) − u ( x , t ) ) d y + f ( u ) ( x , t ) , and a nonlocal analogous to the p-Laplacian, u t ( x , t ) = ∫ R d J ( x , y ) | u ( y , t ) − u ( x , t ) | p − 2 ( u ( y , t ) − u ( x , t ) ) d y . The energy method developed here allows us to obtain decay rates of the form, ‖ u ( ⋅ , t ) ‖ L q ( R d ) ⩽ C t − α , for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/8322926d73c3497e2855a2c2969417ffe29af658
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Fundamental solutions and decay of fully non-local problems (2018, background): “ls with memory. Examples are given by the theory of heat conduction with memory, see, e.g., [26] and the references therein as well as [31], and the diffusion of fluids in porous media with memory, cf. [16]. Given the condition (PC), then (1.1)-(1.2) can be reformulated as an abstract Volterra equatio”