## 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

### Keywords:

quenching; nonlinear boundary condition
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### References:

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