Abstract
Let $(u, v)$ be a solution to a semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_1 \Delta u+v^{p} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ \partial_{t} v = D_2 \Delta v+u^{q} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ u, v \geq 0 \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ (u(\cdot, 0), v(\cdot, 0)) = (\mu, \nu) \quad \mbox{in} \quad \mathbf{R}^N, \end{array} \right. $$ where $N \geq 1$, $T > 0$, $D_1 > 0$, $D_2 > 0$, $0 < p \leq q$ 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
Yohei FUJISHIMA. Kazuhiro ISHIGE. "Initial traces and solvability of Cauchy problem to a semilinear parabolic system." J. Math. Soc. Japan 73 (4) 1187 - 1219, October, 2021. https://doi.org/10.2969/jmsj/84728472
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