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April, 2021 Initial traces and solvability of Cauchy problem to a semilinear parabolic system
Yohei FUJISHIMA, Kazuhiro ISHIGE
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J. Math. Soc. Japan Advance Publication 1-33 (April, 2021). DOI: 10.2969/jmsj/84728472

Abstract

Let $(u, v)$ be a solution to a semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_t u = D_1 \Delta u+v^p \quad \mbox{in} \quad \mathbf{R}^N \times (0,T),\\ \partial_t v = D_2 \Delta v+u^q \quad \mbox{in}\quad \mathbf{R}^N \times (0,T),\\ u,v \ge 0 \quad \mbox{in} \quad \mathbf{R}^N \times (0,T),\\ (u(\cdot,0),v(\cdot,0)) = (\mu,\nu) \quad \mbox{in} \quad \mathbf{R}^N, \end{array} \right. $$ where $N \ge 1$, $T > 0$, $D_1 > 0$, $D_2 > 0$, $0 < p \le q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of Radon measures or nonnegative measurable functions in $\mathbf{R}^N$. In this paper we study qualitative properties of the initial trace of the solution $(u, v)$ and obtain necessary conditions on the initial data $(\mu, \nu)$ for the existence of solutions to problem (P).

Funding Statement

The first author was supported partially by the Grant-in-Aid for Early-Career Scientists (No. 19K14569). The second author of this paper was supported in part by the Grant-in-Aid for Scientific Research (S)(No. 19H05599) from Japan Society for the Promotion of Science.

Citation

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Yohei FUJISHIMA. Kazuhiro ISHIGE. "Initial traces and solvability of Cauchy problem to a semilinear parabolic system." J. Math. Soc. Japan Advance Publication 1 - 33, April, 2021. https://doi.org/10.2969/jmsj/84728472

Information

Received: 27 April 2020; Published: April, 2021
First available in Project Euclid: 4 May 2021

Digital Object Identifier: 10.2969/jmsj/84728472

Subjects:
Primary: 35A01
Secondary: 35K45

Rights: Copyright © 2021 Mathematical Society of Japan

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