Front propagation in nonlinear parabolic equations (Q2922851)

This paper deals with the following Cauchy problem (P) for an unknown function \(u(x,t)\): NEWLINE\[NEWLINE\partial_t u=\text{div}(\partial \Phi(\nabla u))+f(u),\quad u(x,0)=u_o(x),\quad x\in\mathbb R^N, t>0,NEWLINE\]NEWLINE where \(\Phi:\mathbb R^N\to \mathbb R, f:\mathbb R\to \mathbb R, u_o:\mathbb R^N\to\mathbb R\) are suitable given data and \(\partial \Phi\) is the Fréchet derivative of \(\Phi\). In particular, it is assumed that \(\Phi\) is radially symmetric and that there is a constant \(\mu\in (-1,1)\) such that \(f\in C(\mathbb R),f(-1)=f(\mu)=f(1)=0, f<0\text{ in }(-1,\mu),f>0 \text{ in } (\mu,1)\). A standard change of variables leads to the scaled problem \((P_{\epsilon})\): NEWLINE\[NEWLINE\partial_t u_{\epsilon}=\text{div}(\partial \Phi(\epsilon\nabla u_{\epsilon}))+\frac{1}{\epsilon}f(u_{\epsilon}),\quad u_{\epsilon}(x,0)=u_{\epsilon,o}(x),\quad x\in\mathbb R^N, t>0.NEWLINE\]NEWLINE The authors investigate the behavior of solutions \(u_{\epsilon}\) to problem \((P_\epsilon)\) as \(\epsilon\to 0.\) When \(N=1\), setting \(u(x,t)=q(x-ct)\) in (P), where \(c\) is the speed of the wave, leads to the associated one dimensional problem (Q): NEWLINE\[NEWLINEd_x(\partial\Phi (q^{\prime}))+cq^{\prime}+f(q)=0,\quad x\in \mathbb RNEWLINE\]NEWLINE with the conditions NEWLINE\[NEWLINEq^{\prime}\leq 0 \text{ in }\mathbb R,\quad \underset{x\to -\infty}{\lim}q(x)=1,\quad \underset{x\to \infty}{\lim}q(x)=-1, \text{ and } q(0)=\mu.NEWLINE\]NEWLINE They give additional conditions on \(f\) and \(\Phi\) which imply the uniqueness of the solution \((q,c)\) to (Q), such that: NEWLINE\[NEWLINEc=(\int_{\infty}^{\infty}|q^{\prime}|^2 dx)^{-1} \int_{-1}^{1}f(s)ds\geq 0,NEWLINE\]NEWLINE is the set of all. They introduce two regions \(G_{-}\) [resp. \(G_{+}\)] \(x\in\mathbb R^N\) such that there exists a neighbourhood \(U(x)\subset\mathbb R^N\) of \(x\) such that \(\underset{\epsilon\to 0}{\limsup}(\underset{y \in U(x)}{\sup} u_{\epsilon,o}(y))<\mu\), [resp. \(\underset{\epsilon\to 0}{\liminf}(\underset{y \in U(x)}{\inf} u_{\epsilon,o}(y))>\mu ]\), assume \(\bar G_-\cup \bar G_+=\mathbb R^N\) and set \(\Gamma=\bar G_-\cap\bar G_+.\) Assuming, under suitable hypotheses on \(u_{\epsilon,o},\) that \(\{u_{\epsilon}\}\) is a family of admissible weak solutions to (\(P_{\epsilon})\), they prove that, as \(\epsilon\to 0\), \(u_{\epsilon}\to -1\) uniformly in compact subsets of \(\{(x,t): \text{dist}(x,\Gamma)>ct\}\) and \(u_{\epsilon}\to 1\) uniformly in compact subsets of \(\{(x,t): \text{dist}(x,\Gamma)<ct\}\), where dist stands for the signed distance and \(c\) is the unique solution to (Q).