Multinonlinear interactions in quasi-linear reaction-diffusion equations with nonlinear boundary flux
This paper concerns the unique classical solutions of the following initial-boundary value problems with nonlinear boundary flux for the quasilinear reaction-diffusion equations: \[ \begin{aligned} u_t=\Delta\Phi(u)\mp f(u)\quad &\text{in }\Omega\times (0,T),\\ {\partial u\over\partial n}= g(u)\quad &\text{on }\partial\Omega\times (0,T),\\ u(x,0)= u_0(x)\quad &\text{in }\Omega.\end{aligned} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(u_0\) is a positive function, \({\partial u\over\partial n}\) denotes the outward normal derivative of \(u\), \(\Phi'(s)> 0\), which makes the equations strictly parabolic, \(\Phi(s)\), \(f(s)\), \(g(s)\) are nonnegative, and \(\Phi''(s)\geq 0\) for \(s\geq s_0\). The authors prove the existence and nonexistence of the global solutions under some conditions on \(\Phi(s)\), \(f(s)\), \(g(s)\) for \(s\) large. In the case of equation with \(-f(u)\), it is shown that if the absorption \(-f(u)\) is strong enough such that it dominates the system, then the solutions are global \((T=\infty)\). Also, it is shown by constructing blowing up subsolutions that the solutions can blow up in finite time if the corporate role of flux and diffusion is dominant. In the case of equation with \(+f(u)\), under the hypotheses that \(f(s)\), \(g(s)\), \(\Phi'(s)\), \(\Phi'(s)+ \Phi'(s)g'(s)+ \Phi''(s)g(s)\) are positive and nondecreasing, by constructing sub- and supersolutions the authors give the necessary and sufficient condition for the existence of global solution.
scientific article; zbMATH DE number 2082319
scientific article