On the Cauchy problem and initial traces for a degenerate parabolic equation
TL;DR: In this paper, the authors studied the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p−2 Du)(p < 2), and found sufficient conditions on the initial trace u0 (and in particular on its behaviour as |x|→∞) for existence of a solution in some strip RN × (0,T).
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Abstract: The authors study the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p−2 Du)(p<2), and find sufficient conditions on the initial trace u0 (and in particular on its behaviour as |x|→∞) for existence of a solution in some strip RN × (0,T). Using a Harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions. Uniqueness of solutions is shown if u0 belongs to L1loc(RN), but is left open in the case that u0 is merely a locally bounded measure. The results are closely related to papers by Aronson-Caffarelli, Benilan-Crandall-Pierre, and Dahlberg-Kenig about the porous medium equation ut = Δum. The proofs are different and allow one to generalize some of the above results to equations with variable coefficients.
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References
Linear and Quasi-linear Equations of Parabolic Type
O. A. Ladyzhenskai︠a︡,V. A. Solonnikov,V. A. Solonnikov,N. N. Uralʹt︠s︡eva +3 more
- 31 Dec 1968
TL;DR: In this article, the authors introduce a system of linear and quasi-linear equations with principal part in divergence (PCI) in the form of systems of linear, quasilinear and general systems.
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The initial trace of a solution of the porous medium equation
TL;DR: In this paper, the Widder representation theorem was used to prove the existence of continuous weak solutions of the porous medium equation (1.4) whose initial trace is a Borel measure.
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Solutions of the Porous Medium Equation in R(N) under Optimal Conditions on Initial Values.
TL;DR: In this article, the authors established the existence of solutions of the initial value problem under the most general conditions on u(0), i.e., u(t) need only be such that R to the minus (2 divided by m-1 +N) sum (determinant x or = R) to the power of u(x)) dx is bounded independently of R or = 1.
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Regularizing Effects of Homogeneous Evolution Equations
Michael G. Crandall,Philippe Bénilan +1 more
- 01 May 1980
TL;DR: In this article, the authors prove related estimates on nonlinear evolution equations which are governed by homogeneous nonlinearities and apply to classes of nonlinear diffusion equations and to conservation laws.