Global existence and decay estimates for the semilinear heat equation with memory in $\mathbb{R}^{n}$

Document Type : Research Paper


1 Laboratory of Mathematical Analysis, Probability and Optimizations, University of Biskra, Po. Box 145 Biskra (07000), Algeria

2 Department of Mathematics, Mohamed Khider University, B.P. 145, 07000, Biskra, Algeria


In this paper, we study the initial value problem for a semi-linear heat equation with memory in $n$-dimensional space $\mathbb{R}^{n}$. Under a smallness conditions on the initial data, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates in higher Sobolev space of the solution are provided. The proof is carried out by means of the point-wise decay estimates of the solution in the Fourier space and a fixed point-contraction mapping argument.


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Volume 13, Issue 2
July 2022
Pages 2271-2285
  • Receive Date: 20 February 2021
  • Revise Date: 08 March 2021
  • Accept Date: 13 March 2021