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Continuous dependence of the entropy solution of general parabolic equation. (English) Zbl 1122.35012

Summary: We consider the general parabolic equation:
\[ u_t-\Delta b(u)+\text{div\,}F(u)=f\text{ in }Q=]0,T[\times \mathbb R^N, \]
with \(u_0\in L^\infty(\mathbb R^N)\), for a.e \(t\in ]0,T[\), \(f(t)\in L^\infty(\mathbb R^N)\) and \(\int^T_0 \|f(t)\|_{L^\infty(\mathbb R^N)}\,dt<\infty\).
We prove the continuous dependence of the entropy solution with respect to \(F\), \(b\), \(f\) and the initial data \(u_0\) of the associated Cauchy problem.
We start by recalling the definition of weak solution and entropy solution. By applying an abstract result (Theorem 2.3), we get the continuous dependence of the entropy solution. The contribution of the present work consists of considering the equation in the whole space \(\mathbb R^n\) instead of a bounded domain and considering a bounded data instead of integrable data.

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations

Keywords:

bounded data
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