Basic theory for a diffusion-absorption equation in an inhomogeneous medium
The authors study initial-boundary value problems for the parabolic equation \[ \rho(x)u_t = (u ^m)_{xx}-c_0u^p \] for parameters \(m>1\), \(p>0\), and \(c_0>0\) assuming that the function \(\rho\) is positive and bounded from above. They show that the Cauchy-Dirichlet problem has a unique solution under these hypotheses as long as the space-time domain \(\Omega\) on which the problem is posed has the following form \(\{(x,t): \xi _1(t) < x <\xi _2(t), 0<t<T\}\) with \(\xi _1\) either identically \(-\infty\) or a Lipschitz function and \(\xi _2\) either identically \( \infty\) or a Lipschitz function. Hence they can consider problems in non-cylindrical bounded domains, half-bounded domains or all of \(\mathbb R^n\times (0,T)\). The method of proof consists of finding a suitable sequence of approximating problems to which classical existence theory can be applied and then showing that the solutions of the approximating problems satisfy a uniform \(L^\infty\) estimate and converge to a solution of the original problem.
scientific article; zbMATH DE number 1971904