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Existence and asymptotic behavior for a singular parabolic equation. (English) Zbl 1074.35020
The authors firstly show global existence of nonnegative solutions to the singular parabolic equation \[ u_t-\Delta u+\chi _{\left\{u>0\right\}}\left(-u^{-\beta }+\lambda f\left(u \right)\right)=0 \] in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\) with zero Dirichlet boundary condition and initial condition \(u_0\in C\left(\Omega \right)\), \(u_0\geq 0\). Moreover, their results still hold for \(u_0\in L^{\infty}\left(\Omega \right)\) in some cases. Then they show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called “quenching”. Also, they establish a converse of this fact and studied the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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