论文标题
涉及$ p（x）$的抛物线问题 - laplacian，一个力量和单数非线性
A parabolic problem involving $p(x)$-Laplacian, a power and a singular nonlinearity
论文作者
论文摘要
本文的目的是使用$ p（x）$ -laplacian研究非线性奇异抛物线方程。确切地说，我们考虑以下问题，并讨论非负弱解决方案的存在。 \ begin {align*} \ frac {\ partial u} {\ partial t}-Δ__{p（x）} u＆=λu^{q（x）-1} + u^{ - Δ（x）g + f && f && f && f && \ text {in} u＆= 0 && \ text {on} 〜σ_t， u（0，\ cdot）＆= u_0（\ cdot）&& \ text {in} 〜Ω \ nonumber。 \ end {align*} 这里$ q_t =ω\ times（0，t）$，$σ_t= \partialΩ\ times（0，t）$，$ω$是$ \ m \ m i \ mathbb {r}^n $（$ n \ geq 2 $）中的一个有界域，lipschitz contripschitz continum $ \ pottial $ \ partial $ \partialΩ l^1（q_t）$，$ g \ in l^\ infty（ω）$，$ u_0 \ in L^r（ω）$，带有$ r \ geq 2 $，$δ：\overlineΩ\rightlineΩ\rightlineΩ\rightlineΩ $ \ underSet {x \ in \overlineΩ} {\ max} 〜p（x）<n $，$ q（\ cdot）<p^*（\ cdot）$。 根据$ f $的选择，该文章将其分为两种情况。
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=Ω\times(0,T)$, $Σ_T=\partialΩ\times(0,T)$, $Ω$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partialΩ$, $λ\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(Ω)$, $u_0\in L^r(Ω)$ with $r\geq 2$, $δ:\overlineΩ\rightarrow(0,\infty)$ is continuous, and $p,q\in C(\overlineΩ)$ with $\underset{x\in\overlineΩ}{\max}~p(x)<N$, $q(\cdot)<p^*(\cdot)$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot)$, $q(\cdot)$.
