Comparison of solutions of nonlinear evolution problems with different nonlinear terms

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.