Two-parameter asymptotics in a bisingular Cauchy problem for a parabolic equation
The author considers the Cauchy problem for the quasilinear parabolic equation \[ \begin{cases} \frac{\partial u}{\partial t}+\frac{\partial \varphi(u)}{\partial x}=\varepsilon \frac{\partial^2u}{\partial x^2}, & t\geq 0,\quad \varepsilon >0\\ u(x,0,\varepsilon,\rho)= \nu(x\rho^{-1}), & x\in {\mathbb R},\quad \rho>0 \end{cases} \] where \(\varphi \in C^\infty(\mathbb R)\) wiith \(\varphi''>0\) and \(\nu\) is bounded and infinitely differentiable. It is known that there exists a unique bounded infinitely differentiable in \(x\) and \(t\) solution \(u(x,t,\varepsilon,\rho)\) and the paper deals with its asymptotic analysis as \(\varepsilon\to 0\) and \(\rho\to 0.\) The author constructs asymptotic solutions in the form of series in powers of \(\varepsilon\) and \(\rho\) far from the set of discontinuity, and in powers of \(\rho /\varepsilon\) in a small domain of \(x=0,t=0\). The coefficients of this expansion are obtained from a recursive chain of initial value problems.
scientific article; zbMATH DE number 6963752
scientific article