Sharp Estimate of the Life Span of Solutions to the Heat Equation with a Nonlinear Boundary Condition

Abstract

Consider the heat equation with a nonlinear boundary condition

$$\displaystyle \mathrm {(P)}\qquad \left \{ \begin {array}{ll} \partial _t u=\Delta u,\qquad & x\in {\mathbf {R}}^N_+,\,\,\,t>0,\\ \displaystyle {-\frac {\partial u}{\partial x_N} u}=u^p, & x\in \partial {\mathbf {R}}^N_+,\,\,\,t>0,\\ u(x,0)=\kappa \psi (x),\qquad & x\in \overline {{\mathbf {R}}^N_+}, \end {array} \right . \qquad \qquad $$

where N ≥ 1, p > 1, κ > 0 and ψ is a nonnegative measurable function in ${\mathbf {R}}^N_+ :=\{y\in {\mathbf {R}}^N:y_N>0 \}$. Let us denote by T(κψ) the life span of solutions to problem (P). We investigate the relationship between the singularity of ψ at the origin and T(κψ) for sufficiently large κ > 0 and the relationship between the behavior of ψ at the space infinity and T(κψ) for sufficiently small κ > 0. Moreover, we obtain sharp estimates of T(κψ), as κ →∞ or κ → +0.

Appendix

By Theorem 1.1, we obtain Tables 1, 2 and 3. These tables show the behavior of the life span T(κψ) as κ →∞ when ψ is as in Theorem 1.1, that is,

$$\displaystyle \begin{aligned}\psi(x):=|x|{}^{-A}\bigg[\log\left(e+\frac{1}{|x|}\right)\bigg]^{-B}\chi_{B_+(0,1)}(x)\in L^1({\mathbf{R}}^N_+) \setminus L^\infty({\mathbf{R}}^N_+), \end{aligned}$$

where 0 ≤ A ≤ N and

$$\displaystyle \begin{aligned} B>0\quad \mbox{if}\quad A=0, \qquad B\in\mathbf{R}\quad \mbox{if}\quad 0<A<N, \qquad B>1\quad \mbox{if}\quad A=N. \end{aligned}$$

For simplicity of notation, we write T _κ instead of T(κψ).

By Theorem 1.2, we obtain Tables 4 and 5. These tables show the behavior of the life span T(κψ) as κ → +0 when ψ is as in Theorem 1.2, that is, ψ(x) = (1 + |x|)^−A(A > 0).

Table 1 The behavior of T _κ in the case of 1 > p > p _∗ (as κ →∞)

Table 2 The behavior of T _κ in the case of p < p _∗ (as κ →∞)

Table 3 The behavior of T _κ in the case of p = p _∗ (as κ →∞)

Table 4 The behavior of T _κ in the case of A ≠ N (as κ → +0)

Table 5 The behavior of T _κ in the case of A = N (as κ → +0)