论文标题
对于有界域上半线性抛物线方程的全球解决方案存在的新的必要条件
A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on Bounded Domains
论文作者
论文摘要
本文的目的是为以下半连续抛物线方程的全球解决方案的存在和不存在提供必要的条件 \ [ u_ {t} =ΔU+ψ（t）f（u），\，\，\ mbox {in}ω\ times（0，t^{*}）， \] 在有界域上的Dirichlet边界条件下。实际上，几十年来，这一直是一个开放问题，即使对于$ f（u）= u^{p} $的情况。实际上，我们证明： \ [ \ begin {Aligned} ＆\ mbox {且仅在}时，没有任何初始数据的全局解决方案} ＆\ mbox {函数} f \ mbox {满足} ＆\ hspace {20mm} \ int_ {0}^{\ infty}ψ（t）\ frac {f \ left（\ lvert s（t）u_ {0} \ rvert _ {\ rvert _ {\ infty} \ right）} {\ lvert s（t）u_ {0} \ rvert _ {\ infty}} dt = \ infty ＆\ mbox {对于每个} \，ε> 0 \，\ mbox {和非阴性非平底初始数据} \，u_ {0} \ in C_ {0}（ω）。 \ end {Aligned} \] 在这里，$（s（t））_ {t \ geq 0} $是具有Dirichlet边界条件的热量半群。
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 \[ u_{t}=Δu+ψ(t)f(u),\,\,\mbox{ in }Ω\times (0,t^{*}), \] 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: \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if } &\mbox{the function } f \mbox{ satisfies} &\hspace{20mm}\int_{0}^{\infty}ψ(t)\frac{f\left(\lVert S(t)u_{0}\rVert_{\infty}\right)}{\lVert S(t)u_{0}\rVert_{\infty}}dt=\infty &\mbox{for every }\,ε>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(Ω). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition.
