Existence of Compactly Supported Solutions for a Degenerate Nonlinear Parabolic Equation with NonLipschitz Source Term

Abstract

In this paper the authors study the existence of nonnegative compactly supported solutions of a nonlinear degenerate parabolic equation with a non-Lipschitz source term in one space dimension. More precisely, they investigate the properties of nonnegative solutions of the problem {∂tu−ν∂2x(up)=uα+Ψ(u),u(0,x)=u0, x∈R, t>0, u=u(t,x),(1) where Ψ∈C∞ is an increasing function, satisfying Ψ(0)=0 (unforced case), Ψ(x)≤Cx for some C>0 and every x≥0. Equation (1) mimics the properties of the classical k-e system-model in the context of turbulent mixing flows with respect to nonlinearities and support properties of solutions. The authors highlight that the originality of the method resides in the fact that they deal with a non-Lipschitz source term, and in the comparison of not only the speed but also the acceleration of the boundary of the compact support. Here p>1 so that the diffusion is strongly degenerate and α>0 with 2−p≤α<1, so the nonlinearity is not Lipschitz. Equation (1) is a crude simplification of a k-e system of the form ∂tk+∂x(ku)+e=∂x(Cμk2e∂xk)+Pk2e, ∂te+∂x(eu)+C1e2k=∂x(Cμσek2e∂xe)+C2Pk, where k and e denote the specific turbulent kinetic energy and its dissipation rate, u is the velocity field, and P is proportional to turbulence production terms. The present work is focused on the lack of Lipschitz continuity of the source term and the degeneracy of the diffusion. Actually, the study is restricted to the scalar case, as in the k-l turbulence model, where l=k3/2/e is constant. Nonlinear parabolic equations similar to (1) appear also in various applications, most frequently to describe phenomena of thermal propagation in an absorptive medium. Note that such kinds of equations have been studied for slow or fast diffusion and strong or small absorption; see [R. Ferreira and J. L. Vazquez, Nonlinear Anal. 43 (2001), no. 8, Ser. A: Theory Methods, 943-985; MR1812069 (2002f:35141); R. Ferreira, V. A. Galaktionov and J. L. Vazquez, Nonlinear Anal. 50 (2002), no. 4, Ser. A: Theory Methods, 495-507; MR1923525 (2003h:35109); V. A. Galaktionov, S. I. Shmarev and J. L. Vazquez, Arch. Ration. Mech. Anal. 149 (1999), no. 3, 183-212; MR1726675 (2001k:35167)] and the references therein. Finally, the authors prove the following result: Theorem. Assume that u0 is a smooth nonnegative function with support of the form [α,β] and that the derivatives of up0 at α, β do not vanish. Then there exists a smooth, nonnegative, compactly supported solution u to (1) defined for all positive times.