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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
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