Nonlinear parabolic equations with critical initial conditions $$J(u_0)=d$$ or $$I(u_0)=0$$.(English)Zbl 1059.35064

Summary: We study the initial boundary value problem of nonlinear parabolic equation \begin{aligned} \frac{\partial u}{\partial t} =\sum^N_{i=1} \frac {\partial} {\partial x_i}\left(\left|\frac{\partial u}{\partial x_i} \right|^{p-2} \frac{\partial u}{\partial x_i}\right)+u^{1+\alpha}, \quad & x\in\Omega,\;t>0,\\ u(x,0)=u_0(x),\quad & x\in\Omega,\\ u(x,t)=0,\quad & x\in\partial\Omega,\;t\geq 0,\end{aligned} where $$\Omega \subset\mathbb{R}^N$$ is a bounded domain, $$p<2+\alpha< \infty$$ if $$p\geq N$$; $$p>2+\alpha<\frac{Np}{N-p}$$ if $$p<N$$. By using potential well methods we prove that if $$0\leq u_0(x)\in W_0^{1,p}(\Omega)$$, $$J(u_0)=d$$, $$I(u_0)>0$$ or $$I(u_0)=0$$, $$0<J(u_0)\leq d$$, then the problem admits a global solution $$u(t)\leq L^\infty(0,\infty;W_0^{1,p} (\Omega))$$ with $$u_t(t)\in L^2(0, \infty;L^2(\Omega))$$ and $$u(t)\in \widetilde W$$ for $$0\leq t<\infty$$.

MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K65 Degenerate parabolic equations

Keywords:

potential well; Global solution
Full Text:

References:

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