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Resumen:
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The authors study the nonlinear porous media type equation ut(t,x)???(u(t,x))=0 for (t,x)?(0,?)×?, ?(u(t,x))=0 for (t,x)?(0,?)×??, u(0,x)=u0(x) for x??, with ? an open set in Rn, and ? a regular, real, continuous, nondecreasing function. In the classical framework, the following theorem is proved: Let ?i?C2(R) with ?i?>0 and u0i?C(?¯¯¯)?L?(?), for i=1,2. Then if (i) ?1(u01)??2(u02) on ?, (ii) ??1???2 on R, where ?i=??1i, and (iii) ??2(u02)?0 on ?, we have ?1(u1)??2(u2) on (0,?)×?. A counterexample shows the necessity of (iii). The theorem is proved by an application of the maximum principle. In a more abstract framework, a similar theorem is proved for the abstract Cauchy problem du/dt+Au?f, u(0)=u0, where A operates as a multiapplication in a Banach space X, u0?X, and f?L1(0,T:X). The abstract result is applied to well-posed Cauchy problems in L1(?). Generalizations are given, including nonlinear boundary conditions and replacing the Laplacian operator ? by a generalized (nonlinear) Laplacian.
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