论文标题
（P，Q）单数抛物线方程的定性研究：局部存在，Sobolev规律性和渐近行为
A qualitative study of (p,q) Singular parabolic equations: local existence, Sobolev regularity and asymptotic behaviour
论文作者
论文摘要
本文的目的是研究以下抛物面$（p，q）$ - 单数方程的存在，规律性，稳定和爆炸结果： \ begin {equation*} （p_t）\; \ left \ {\ begin {array} {rllll} u_t -δ_{p} u-Δ__{q} u＆= \ vth \; u^{ - \ de}+ f（x，u），\; u> 0 \ text {in} \ om \ times（0，t），\\ u＆= 0 \ quad \ text {on} \ pa \ om \ times（0，t）， u（x，0）＆= u_0（x）\; \ text {in} \ om， \ end {array} \正确的。 \ end {equation*} 其中$ \ om $是$ \ mathbb {r}^n $中的一个有限域，带有$ c^2 $ boundard $ \ pa \ om $，$ 1 <q <q <q <q <p <p <\ infty $，$ 0 <\ de，t> 0 $，$ n $，$ n \ ge 2 $ 2 $ and $ \ vth> 0 $是参数。此外，我们假设$ f：\ om \ times [0，\ infty）\ to \ mb r $在carathéodory函数下方是一个界限，本地lipschitz相对于第二个变量，在$ x \ in \ om $中均匀地均匀地属于\ om $和$ u_0 \ in l^\ of infty（\ om om）\ om cap w^^^^^^^^^^^^^1，p}根据$ f $的增长，我们将这些案例区分为$ q $ - subhomosous和$ q $ - superhomenyoos（以下是我们将丢弃$ q $）。在下均匀的情况下，我们证明了$ \ de <2+1/（p-1）$的问题$（p_t）$的弱解决方案的存在和独特性。为此，我们首先通过使用子和超级解决方案的方法研究了与$（p_t）$相对应的固定问题，并随后采用了隐式欧拉方法，我们获得了$（p_t）$的解决方案。此外，在这种情况下，我们证明了稳定结果，即$（p_t）$的解决方案$ u（t）$（p_t）$收敛到$ u_ \ infty $，$ l^\ infty（\ om）$ as $ t \ ra \ ra \ ra \ infty $。对于超均匀案例，我们通过帮助非线性半群理论来证明局部存在定理。随后，我们证明有限的时间吹入问题$（p_t）$的小参数$ \ vartheta> 0 $在情况下$ \ de \ leq 1 $，以及所有$ \ vth> 0 $的情况，在情况下为$ \ de> 1 $。
The purpose of the article is to study the existence, regularity, stabilization and blow up results of weak solution to the following parabolic $(p,q)$-singular equation: \begin{equation*} (P_t)\; \left\{\begin{array}{rllll} u_t-Δ_{p}u -Δ_{q}u & = \vth \; u^{-\de}+ f(x,u), \; u>0 \text{ in } \Om\times (0,T), \\ u&=0 \quad \text{ on } \pa\Om\times (0,T), u(x,0)&= u_0(x) \; \text{ in }\Om, \end{array} \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1<q<p< \infty$, $0<\de, T>0$, $N\ge 2$ and $\vth>0$ is a parameter. Moreover, we assume that $f:\Om\times [0,\infty) \to \mb R$ is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in $x\in\Om$ and $u_0\in L^\infty(\Om)\cap W^{1,p}_0(\Om)$. We distinguish the cases as $q$-subhomogeneous and $q$-superhomogeneous depending on the growth of $f$ (hereafter we will drop the term $q$). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem $(P_t)$ for $\de<2+1/(p-1)$. For this, we first study the stationary problems corresponding to $(P_t)$ by using the method of sub and super solutions and subsequently employing implicit Euler method, we obtain the existence of a solution to $(P_t)$. Furthermore, in this case, we prove the stabilization result, that is, the solution $u(t)$ of $(P_t)$ converges to $u_\infty$, the unique solution to the stationary problem, in $L^\infty(\Om)$ as $t\ra\infty$. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow up of solution to problem $(P_t)$ for small parameter $\vartheta>0$ in the case $\de\leq 1$ and for all $\vth>0$ in the case $\de>1$.
