Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

¹ Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam.
² Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam.
³ Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India.
⁴ Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam.

^* Corresponding author: nguyenhuytuan@tdmu.edu.vn

Received: 15 June 2020
Accepted: 28 February 2021

Abstract

In this paper, we investigate a initial value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace operator of order 0 < ν ≤ 1 and the nonlinear memory source term. For 0 < ν < 1, the problem will be considered on a bounded domain of ℝ^d. By some Sobolev embeddings and the properties of the Mittag-Leffler function, we will give some results on the existence and the uniqueness of mild solution for problem (1.1) below. When ν = 1, we will introduce some L^p − L^q estimates, and based on them we derive the global existence of a mild solution in the whole space ℝ^d.