A relaxation approximation for degenerate parabolic equations
The author considers the Cauchy problem for the parabolic equation \[ u_t+ \sum^d_{j= 0} (A_j(u))_{x_j}- \Delta(B(u))= G(u),\quad u(x,0)= u_0(x),\tag{1} \] where \(x\in \mathbb{R}^d\), \(u_0\in L^1(\mathbb{R}^d)\cap L^\infty(\mathbb{R}^d)\) and \(A_j\), \(B\) and \(C\) are smooth functions with \(B'(u)\geq 0\). He introduces the following semilinear system of equations \[ \begin{aligned} \partial_t f^\varepsilon_i+ \sum^d_{j=1} \lambda^i_j\partial x_j f^\varepsilon_i- \mu^i\Delta f^\varepsilon_i & = {1\over \varepsilon}(M_i(u^\varepsilon)- f^\varepsilon_i)+ \beta_i G(u^\varepsilon),\\ f^\varepsilon_i(x, 0) & = f^\varepsilon_{0i}(x),\;i= 1,2,\dots, N,\end{aligned}\tag{2} \] that approximates system (1) in the sense that \(u^\varepsilon(x, t)= \sum^N_{i=1} f^\varepsilon_i(x, t)\) approximates the solution \(u\) of problem (1). At first the author proves the existence, uniqueness and \(L^1\)-stability of the solutions of system (2). Using the obtained estimates, he proves that there exists a subsequence \(\{f^{\varepsilon_k}_i\}\) that converges to \(f_i\), when \(\varepsilon_k\) goes to zero, and thus \(u^\varepsilon(x, t)@>>\varepsilon\to 0> u(x,t)= \sum^n_{i=1} f_i(x,t)\), where \(u(x,t)\) is the solution of problem (1). At the end, four illustrating examples are presented.
scientific article; zbMATH DE number 1513207