Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations
The paper deals with the problem in what way the solution to the parabolic equation \[ \partial_t v=\nabla\bigl(\Phi (v)\bigr)+\Delta \bigl( \varphi(v)\bigr) \] with the initial condition \(v(0,\cdot)=h\) in the domain \((0, \infty)\times\mathbb{R}^d\) depends on functions \(\Phi\) and \(\varphi\). These functions are uniformly differentiable, moreover, \(\varphi\) is nondecreasing. Basing on the properties of nonlinear semigroups, the authors prove that the solution to the initial problem uniformly depends on the initial conditions \(h\) and the functions \(\Phi\) and \(\varphi\). To be more precise, if \(v_j\), \(j=1,2\), satisfy \[ \partial_tv_j= \nabla\bigl(\Phi_j (v_j)\bigr)+ \Delta\bigl(\varphi_j (v_j) \bigr), \quad v_j(0,\cdot) =k_j, \] then \[ \begin{multlined}\bigl\| v_1(t,\cdot)-v_2 (t,\cdot)\bigr\|_{L^1(\mathbb{R}^d)} \leq\| h_1-h_2\|_{L^2 (\mathbb{R}^d)}+\| h_1 \|_{TV(\mathbb{R}^d)} \times\\ \times\Bigl(t\sup_{s\in I(h_1)}\bigl\|\Phi_1'(s)-\Phi_2'(s)\bigr\|_\infty +4\sqrt{td}\sup_{s\in I(h_1)}\bigl| \sqrt {\varphi_1'(s)}- \sqrt{\varphi_2'(s)} \bigr|\Bigr). \end{multlined} \] where \(I(h) {\overset{def} =}-\|h^-\|_{L^\infty},\bigl(\|h^+\|_{L^\infty}\bigr)\).
scientific article; zbMATH DE number 1271347