Continuous dependence of renormalized solution for nonlinear degenerate parabolic problems in the whole space
The authors study the continuous dependence of the so-called renormalized solution for the general degenerate parabolic equation \[ u_t - \Delta\,b(u) + \operatorname{div}F(b(U)) = f \qquad \text{in } Q = (0,T)\times\mathbb R^N,\, T > 0, \] under the initial condition \[ u(0,\cdot) = u_0 \qquad \text{on }\mathbb R^N, \] where \(F:\mathbb R \to\mathbb R^N\) is a Lipschitz continuous function and \(b\) is a nondecreasing continuous function on \(\mathbb R\). For normalization, the values \(b(0) = 0\) and \(F(0) = 0\) are fixed. The well-posedness of the renormanized solution for this problem with \(L^1\) initial data was studied in [\textit{M. Maliki} and \textit{A. Ouedraogo}, Ann. Fac. Sci. Toulouse, Math. (6) 17, No. 3, 597--611 (2008; Zbl 1173.35565)]. The main aim of this paper is to establish the continuous dependence of the renormalized solution with respect to the data. The novelty is that the authors work in the whole space \(\mathbb R^N\) with unbounded data \((u_0,f)\).
scientific article; zbMATH DE number 6358076
scientific article