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arXiv 1104.1691math.AP
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一类奇异抛物型方程：解的存在性和稳定性

A Singular Parabolic Equation: Existence, Stabilization

Mehdi Badra, Kaushik Bal, Jacques Giacomoni

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31 pages
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摘要：

研究了拟线性抛物奇异方程\tag{\rm P$_t$} \{{aligned} & u_t-Δ_p u = \frac{1}{u^δ} +f（x，u）\; \text{in} \，（0，T）\times Ω，& u =0\，\text{on} \;（0，T）\times \partial Ω，\quad u>0 \text{in} \，（0，T）\times Ω，&u（0，x）=u_0（x）\; \text{in} Ω，{aligned}.其中$Ω$是在$\R^{\rm N}$、$1 < p< \infty$、$0<δ$和$T>0$中具有光滑边界的开有界域。我们假设$(x,s)\inΩ\times\R^+\to f(x,s)$是一个下有界Caratheodory函数，局部Lipschitz关于$s$在$x\inΩ$中一致且渐近次齐次，即% {方程} \label{sublineargrowth} 0 \leq \displaystyle \lim_{t\to +\infty} \frac{f(x,t)}{t^{p-1}} =α_f<λ_1（Ω），{方程} %（其中$λ_1(Ω)$是具有齐次Dirichlet边界条件的$Ω$中$-Δ_p$的第一特征值）和$u_0\in L^\infty(Ω)\cap W^{1,p}_0(Ω)$，满足下面定义的锥条件。然后，对任意的$δ\in (0,2+\frac{1}{p-1})$，证明了$u \in{\bf V}(Q_T)$到$({\rm P_t})$的弱解的存在唯一性.此外，$u\in C([0,T], W^{1,p}_0(Ω))$和限制$δ<2+\frac{1}{p-1}$是尖锐的。最后，在最后一节中，我们分析了$p=2$案例。利用插值空间理论和半群理论，证明了在$C([0,T], L^2(Ω))\cap L^\infty(Q_T)$中任意$δ>0$的$({\rm P}_t)$弱解的存在性和唯一性，并在适当的初始条件下，给出了一些正则性结果.最后，我们描述了他们的渐近行为在$L^\infty(Ω)\cap H^1_0(Ω)$时$δ<3$。

英文摘要：

We investigate the following quasilinear parabolic and singular equation, {equation} \tag{\rm P$_t$} \{{aligned} & u_t-Δ_p u =\frac{1}{u^δ}+f(x,u)\;\text{in}\,(0,T)\timesΩ, & u =0\,\text{on} \;(0,T)\times\partialΩ,\quad u>0 \text{in}\, (0,T)\timesΩ, &u(0,x) =u_0(x)\;\text{in}Ω, {aligned}. {equation} % where $Ω$ is an open bounded domain with smooth boundary in $\R^{\rm N}$, $1 < p< \infty$, $0<δ$ and $T>0$. We assume that $(x,s)\inΩ\times\R^+\to f(x,s)$ is a bounded below Caratheodory function, locally Lipschitz with respect to $s$ uniformly in $x\inΩ$ and asymptotically sub-homogeneous, i.e. % {equation} \label{sublineargrowth} 0 \leq\displaystyle\lim_{t\to +\infty}\frac{f(x,t)}{t^{p-1}}=α_f<λ_1(Ω), {equation} % (where $λ_1(Ω)$ is the first eigenvalue of $-Δ_p$ in $Ω$ with homogeneous Dirichlet boundary conditions) and $u_0\in L^\infty(Ω)\cap W^{1,p}_0(Ω)$, satisfying a cone condition defined below. Then, for any $δ\in (0,2+\frac{1}{p-1})$, we prove the existence and the uniqueness of a weak solution $u \in{\bf V}(Q_T)$ to $({\rm P_t})$. Furthermore, $u\in C([0,T], W^{1,p}_0(Ω))$ and the restriction $δ<2+\frac{1}{p-1}$ is sharp. Finally, in the last section we analyse the case $p=2$. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to $({\rm P}_t)$ for any $δ>0$ in $C([0,T], L^2(Ω))\cap L^\infty(Q_T)$ and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in $L^\infty(Ω)\cap H^1_0(Ω)$ when $δ<3$.

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