Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes

For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.     

Weighted estimates of solutions to second order parabolic equations

We evaluate the rate of decay for solutions to second order parabolic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. Our approach is based on a special growth lemma, and it works for both divergence and non-divergence equations, in domains satisfying a general “exterior measure condition” (A). The result for elliptic case is published in Cho and Safonov (2007) [2].     

Harnack estimates for a quasi-linear parabolic equation with a singular weight

In this paper, based on measure theoretical arguments, we establish Harnack estimates and Hölder continuity of nonnegative weak solutions for a degenerate parabolic equation with a singular weight. We transform the equation by performing the change of function. The energy estimates, the upper boundedness, the lower boundedness and the expansion of positivity for the solutions to the transformed equation are obtained. Then our aim is reached.     

Regularity for the porous medium equation with variable exponent: The singular case

We extend to the singular case the results of [E. Henriques, J.M. Urbano, Intrinsic scaling for PDEs with an exponential nonlinearity, Indiana Univ. Math. J. 55 (5) (2006) 1701-1721] concerning the regularity of weak solutions of the porous medium equation with variable exponent. The method of intrinsic scaling is used to show that local weak solutions are locally continuous.     

Well-posedness results for triply nonlinear degenerate parabolic equations

We study well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problems of the kind

     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

Singular and degenerate anisotropic parabolic equations with a nonlinear source

The Dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered. We state conditions that guarantee the existence and uniqueness of a global weak solution to the problem. A similar result is proved for the parabolic p-Laplace equation.     

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose L^∞$L^{\infty }$-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique.     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

Obstacle problem for nonlinear parabolic equations

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.     

Lower semicontinuous obstacles for the porous medium equation

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.     

On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity

(P)

     

On some global well-posedness and asymptotic results for quasilinear parabolic equations

We prove that the quasilinear parabolic initial-boundary value problem (1.1) below is globally well-posed in a class of high order Sobolev solutions, and that these solutions possess compact, regular attractors ast+.     

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.