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Quenching on the boundary. (English) Zbl 0809.35043

Let \(u\) be the solution of the equation \(u_ t= u_{xx}\), \(x\in (0,1)\), \(t>0\), with the boundary conditions \(u_ x (0,t)=0\), \(u_ x(1,t)= - u^{-\beta} (1,t)\), where \(\beta>0\), and the initial condition \(u_ 0\) is positive, sufficiently smooth and satisfies the boundary conditions. The authors show that this solution reaches zero at \(x=1\) in finite time \(T=T(u_ 0)\). Concerning the behavior of \(u\) near \((1,T)\), they prove the following estimates: \[ \begin{aligned} C_ 1\leq (1-x)^{-2\lambda} u(x,T)\leq C_ 2 &\qquad \text{for \(x\) close to } 1\\ (T-t)^{-\lambda} u(1-y \sqrt{T-t}, t)\to z_ 0(y) &\qquad \text{as }t\to T-,\end{aligned} \] where \(z_ 0\) is a solution of an O.D.E. in \(\mathbb{R}^ +\) and \(\lambda= 1/(2\beta +2)\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
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References:

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