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- W4286229603 abstract "The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations [ u_{t}=Delta u+psi(t)f(u),,,mbox{ in }Omegatimes (0,t^{*}), ] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove: [ begin{aligned} &mbox{there is no global solution for any initial data if and only if } &mbox{the function } f mbox{ satisfies} &hspace{20mm}int_{0}^{infty}psi(t)frac{fleft(lVert S(t)u_{0}rVert_{infty}right)}{lVert S(t)u_{0}rVert_{infty}}dt=infty &mbox{for every },epsilon>0,mbox{ and nonnegative nontrivial initial data },u_{0}in C_{0}(Omega). end{aligned} ] Here, $(S(t))_{tgeq 0}$ is the heat semigroup with the Dirichlet boundary condition." @default.
- W4286229603 created "2022-07-21" @default.
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- W4286229603 date "2022-07-18" @default.
- W4286229603 modified "2023-10-16" @default.
- W4286229603 title "A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on Bounded Domains" @default.
- W4286229603 doi "https://doi.org/10.48550/arxiv.2207.08383" @default.
- W4286229603 hasPublicationYear "2022" @default.
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