ABSTRACT. We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following:in Ω,

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equationsThese two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771(Phys. D, 240, -1784(Phys. D, 240, , 2011, respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.

We study a class of parabolic equations having first-order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.

In this paper we study the large time behavior of the solutions to the following nonlinear fourthorder equationsThese two PDE were proposed as models of the evolution of crystal surfaces by J.

In this paper we deal with uniqueness of solutions to the following problemthe parabolic cylinder, Ω is an open subset of R N , N ≥ 2, 1 < p < N , and the right hand side H(t, x, ξ) : (0, T ) × Ω × R N → R exhibits a superlinear growth with respect to the gradient term. 2010 Mathematics Subject Classification. 35B51, 35K55,35K61.