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Comparison of solutions of nonlinear evolution problems with different nonlinear terms

Díaz Díaz, Jesús Ildefonso and Benilan, Philippe (1982) Comparison of solutions of nonlinear evolution problems with different nonlinear terms. Israel Journal of Mathematics , 42 (3). pp. 241-257. ISSN 0021-2172

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Abstract

The authors study the nonlinear porous media type equation ut(t,x)−Δφ(u(t,x))=0 for (t,x)∈(0,∞)×Ω, φ(u(t,x))=0 for (t,x)∈(0,∞)×∂Ω, u(0,x)=u0(x) for x∈Ω, with Ω an open set in Rn, and φ a regular, real, continuous, nondecreasing function. In the classical framework, the following theorem is proved: Let φi∈C2(R) with φi′>0 and u0i∈C(Ω¯¯¯)∩L∞(Ω), for i=1,2. Then if (i) φ1(u01)≤φ2(u02) on Ω, (ii) ψ′1≤ψ′2 on R, where ψi=φ−1i, and (iii) Δφ2(u02)≤0 on Ω, we have φ1(u1)≤φ2(u2) on (0,∞)×Ω. 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(Ω). Generalizations are given, including nonlinear boundary conditions and replacing the Laplacian operator Δ by a generalized (nonlinear) Laplacian.

Item Type:Article
Uncontrolled Keywords:nonlinear evolution problems; porous media; abstract Cauchy problems
Subjects:Sciences > Mathematics > Differential equations
ID Code:16437
Deposited On:19 Sep 2012 08:01
Last Modified:19 Sep 2012 08:01

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