On the semigroup approach to a class of space-dependent porous medium systems (Q1772544)

The article is devoted to the study of the following initial-boundary value problem for nonlinear systems of porous medium equations \[ \begin{aligned} &u_{1t} = \Lambda\phi_1(x,u_1) + f_1(x,u_2,u_1),\\ &u_{2t} = \Lambda\phi_2(x,u_1) + f_2(x,u_2,u_1),\\ &\left.u_1(\cdot,t)\right| _{\partial\Omega} = \left.u_2(\cdot,t)\right| _{\partial\Omega} = 0,\quad t\geq 0,\\ &u_1(x,0) = u_1^0,\quad u_2(x,0) = u_2^0,\quad u_1^0, u_2^0 \in L^{\infty}(\Omega). \end{aligned} \] Here \(\Omega\) is a bounded domain in \(\mathbb R^n\), and it is assumed that \(\partial\Omega\) is sufficiently smooth. First, for an auxiliary initial-boundary value problem for a single equation of the above-mentioned type the existence of solutions is proven by using nonlinear semigroup theory. Second, the authors extend the result for the single equation to the original problem and show that the conditions of the semigroup generation theorem are satisfied too. As a result, unique solutions in the sense of distribution are constructed.