The asymptotic behavior of solution for the singularly perturbed initial boundary value problems of the reaction-diffusion equations in a part of domain. (Q1596409)

The authors deal with the following initial boundary value problem for reaction-diffusion equations of the form \[ \begin{cases} u_t-\lambda_\varepsilon (x)\bigl(\mu(u)u_x\bigr)_x+K_x(u)+f(x,t,u)=0\\ u(t,0)= u(t,1)=0,\;u(0,x)=0\end{cases} \tag{1} \] where \((t,x)\in(0,T)\times ((0,\alpha)\cup (\alpha,1))\), \(\varepsilon\) is a small positive parameter, \(\mu,K,f\) are differentiable functions, \(K\) is strictly monotone function in \(u\), \(\mu(u)\geq \mu_0>0\), \(f_u'\geq C_0>0\) and \(\alpha\in(0,1)\) is a constant. Here \(\lambda_\varepsilon (x)=1\) if \(x\in(0,\alpha]\), and \(=\varepsilon\) if \(x\in(\alpha,1)\). Using the operator theory the authors study the asymptotic behaviour of solutions of (1).