Some basic properties of bounded solutions of parabolic equations with p-Laplacian diffusion (original) (raw)
Global solvability results for parabolic equations with p-Laplacian type diffusion
Journal of Mathematical Analysis and Applications, 2018
We give conditions that assure global existence of bounded weak solutions to the Cauchy problem of general conservative 2nd-order parabolic equations with p-Laplacian type diffusion (p > 2) and initial data u 0 ∈ L 1 (R n) ∩ L ∞ (R n). Related results of interest are also given.
Very weak solutions of parabolic systems of p-Laplacian type
Arkiv för Matematik, 2002
We show that the standard assumptions on weak solutions to certain parabolic systems can be weakened and still the usual regularity properties of solutions can be obtained. In order to do this, we derive estimates for the solutions below the natural exponent and then apply reverse HSlder inequalities. Our work is motivated by the classical Weyl's lemma: If a locally integrable function satisfies Laplace's equation in the sense of distributions, t h e n it is real analytic. In other words, only a very modest requirement on the regularity of a solution is needed for a partial differential equation to make sense and t h e n the equation gives extra regularity. We are interested in nonlinear parabolic systems of partial differential equations so t h a t a counterpart of Weyl's l e m m a is too much to hope for, b u t the question of relaxing the s t a n d a r d Sobolev type assumptions on weak solutions and still o b t a i n i n g regularity theory is the objective of our work. We consider solutions to second order parabolic systems OUi=divAi(x,t, Vu)+Bi(x, L Vu), i=l,...,N. (1.1) at In particular, we are interested in systems of p -L a p l a c i a n type. The principal prototype is the p-parabolic system COUi _div(lV~tlp_2Vui)
A Boundary Estimate for Degenerate Parabolic Diffusion Equations
Potential Analysis
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of p-laplacian type. The estimate is given in terms of a Wienertype integral, defined by a proper elliptic p-capacity.
A boundary estimate for singular parabolic diffusion equations
Nonlinear Differential Equations and Applications NoDEA
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of p-laplacian type. The estimate is given in terms of a Wienertype integral, defined by a proper elliptic p-capacity.
NONUNIQUENESS OF SOLUTIONS OF INITIAL-VALUE PROBLEMS FOR PARABOLIC p-LAPLACIAN
Electronic Journal of Differential Equations, 2015
We construct a positive solution to a quasilinear parabolic problem in a bounded spatial domain with the p-Laplacian and a nonsmooth reaction function. We obtain nonuniqueness for zero initial data. Our method is based on sub-and supersolutions and the weak comparison principle. Using the method of sub-and supersolutions we construct a positive solution to a quasilinear parabolic problem with the p-Laplacian and a reaction function that is non-Lipschitz on a part of the spatial domain. Thereby we obtain nonuniqueness for zero initial data.
2018
We discuss comparison principles, the asymptotic behaviour, and the occurrence of blow up phenomena for nonlinear parabolic problems involving the p–Laplacian operator of the form ∂tu = ∆pu + f (t,x,u) in Ω for t > 0, σ∂tu + |∇u|p−2∂νu = 0 on ∂Ω for t > 0, u(0, ·) = u0 in Ω, where Ω is a bounded domain of RN with Lipschitz boundary, and where ∆pu := div ( |∇u|p−2∇u ) is the p–Laplacian operator for p > 1. As for the dynamical time lateral boundary condition σ∂tu + |∇u|p−2∂νu = 0 the coefficient σ is assumed to be a nonnegative constant. In particular, the asymptotic behaviour in the large for the parameter dependent nonlinearity f (·, ·,u) = λ|u|q−2u will be investigated by means of the evolution of associated norms.
Gradient Bounds for Solutions of Elliptic and Parabolic Equations
Lecture Notes in Pure and Applied Mathematics, 2005
Let L be a second order elliptic operator on R d with a constant diffusion matrix and a dissipative (in a weak sense) drift b ∈ L p loc with some p > d. We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure µ satisfying the equation L * µ = 0 and that the closure of L in L 1 (µ) generates a Markov semigroup {T t } t≥0 with the resolvent {G λ } λ>0 . We prove that, for any Lipschitzian function f ∈ L 1 (µ) and all t, λ > 0, the functions T t f and G λ f are Lipschitzian and