论文标题
解决抛物线问题的解决方案的稳定性，涉及分数p-laplacian具有对数非线性的稳定性
Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity
论文作者
论文摘要
在本文中，我们研究了涉及分数$ p $ laplacian具有对数非线性的抛物线方程的以下Dirichlet问题 \ begin {qore*} \ label {eq} \ left \ { \ begin {array} {llc} u_ {t}+（ - δ）^{s} _ {p} u+| u | u |^{p-2} u = | u | u |^{p-2} u \ log（| u |）＆\ text {in} u = 0＆\ text {in}＆\ mathbb {r}^{n} \backslashΩ，\; t> 0， u（x，0）= u_ {0}（x），＆\ text {in}＆ω， \ end {array} \ right。 \ end {equation*}其中$ω\ subset \ mathbb {r}^n \，（n \ geq 1）$是一个带有Lipschitz边界的有界域，$ 2 \ leq p <\ iffty $。局部存在将通过使用Galerkin近似值来完成。通过将潜在的井理论与Nehari流形相结合，我们确定了全球解决方案的存在。然后，借助差异不平等技术，我们证明了局部解决方案在有限时间内具有任意负初始能量和合适的初始值。此外，我们给出了全球解决方案的衰减估计。这里的主要困难是缺乏关于分数$ p $ -laplacian的对数Sobolev不平等。
In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional $p$-Laplacian with logarithmic nonlinearity \begin{equation*}\label{eq}\left\{ \begin{array}{llc} u_{t}+(-Δ)^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log(|u|) & \text{in}\ & Ω,\;t>0 , u =0 & \text{in} & \mathbb{R}^{N}\backslash Ω,\;t > 0, u(x,0)=u_{0}(x), & \text{in} &Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^N \, ( N\geq 1)$ is a bounded domain with Lipschitz boundary and $2\leq p< \infty$. The local existence will be done by using the Galerkin approximations. By combining the potential well theory with the Nehari manifold we establish the existence of global solutions. Then, by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional $p$-Laplacian.
