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Quenching on the boundary. (English) Zbl 0809.35043
Let $u$ be the solution of the equation $u\sb t= u\sb{xx}$, $x\in (0,1)$, $t>0$, with the boundary conditions $u\sb x (0,t)=0$, $u\sb x(1,t)= - u\sp{-\beta} (1,t)$, where $\beta>0$, and the initial condition $u\sb 0$ is positive, sufficiently smooth and satisfies the boundary conditions. The authors show that this solution reaches zero at $x=1$ in finite time $T=T(u\sb 0)$. Concerning the behavior of $u$ near $(1,T)$, they prove the following estimates: $$\align C\sb 1\leq (1-x)\sp{-2\lambda} u(x,T)\leq C\sb 2 &\qquad \text{for $x$ close to } 1\\ (T-t)\sp{-\lambda} u(1-y \sqrt{T-t}, t)\to z\sb 0(y) &\qquad \text{as }t\to T-,\endalign$$ where $z\sb 0$ is a solution of an O.D.E. in $\bbfR\sp +$ and $\lambda= 1/(2\beta +2)$.

35K60Nonlinear initial value problems for linear parabolic equations
35B40Asymptotic behavior of solutions of PDE
35K05Heat equation
Full Text: DOI
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