Continuous dependence of generalized local solutions. (Dépendance continue de solutions généralisées locales.) (French. English summary) Zbl 1029.35027

Summary: We consider the general parabolic equation \[ u_t+f(u)_x =\beta(u)_{xx}+ v\quad\text{in }Q=] 0,T[\times\mathbb{R},\;T>0. \] We prove the continuous dependence of the local generalized solution with respect to \(f, \beta,v\) and the initial data \(u_0\) of the associated Cauchy problem. This type of solution was introduced and studied in the author’s paper [ibid. 7, 113-133 (1998; Zbl 0914.35068)]. We start by recalling different properties of the local generalized solution. We give an abstract general lemma and, in application of this result we get the continuous dependence of local generalized solutions.


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


Zbl 0914.35068
Full Text: DOI Numdam EuDML


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