GLOBAL WELL-POSEDNESS OF A CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION WITH MEMORY

Anh Tuan Nguyen, Bui Dai Nghia and Nguyen van Thinh
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Anh Tuan Nguyen: Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam2Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam
Bui Dai Nghia: Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam4Vietnam National University, Ho Chi Minh City, Vietnam5Department of Mathematics, Faculty of Science, Nong Lam University, Ho Chi Minh City, Vietnam
Nguyen van Thinh: Department of Civil and Environmental Engineering, Seoul National University, Republic of Korea

FRACTALS (fractals), 2023, vol. 31, issue 04, 1-11

Abstract: In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources.

Keywords: Fractional Derivative; Nonlinear Parabolic Equation; Exponential Source; Memory Effects (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23400133

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