Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions degenerate nonlinear parabolic equation with initial data being sum periodic function and vanishing at infinity in appropriate sense.

In this paper, we establish the local existence and uniqueness of the solution for the degenerate parabolic equation with a nonlocal source and homogeneous Dirichlet boundary condition. Moreover, we prove that the solution blows up in finite time and obtain the blow-up set in some special case. Mathematics Subject Classification: 35K20, 35K30, 35K65

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops L∞ x -norm blowup in finite time.

Using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain. We assume a polynomial type growth on the nonlinearity, and an exponential growth on the external force. The obtained results extend some existing results for non-autonomous reaction-diffusi...

Abstract. Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method ...

We study the existence and long-time behavior of solutions to a semilinear strongly degenerate parabolic equation on R under an arbitrary polynomial growth order of the nonlinearity. To overcome some significant difficulty caused by the lack of compactness of the embeddings, the existence of global attractors is proved by combining the tail estimates method and the asymptotic a priori estimate ...

We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance: ∂xxu+ u∂yu− ∂tu = f. Using a harmonic analysis technique on Lie groups, we prove that, if the solution u satisfies condition ∂xu 6= 0 in an open set Ω ⊂ R and f ∈ C∞(Ω), then u ∈ C∞(Ω).

In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynam...

An existence result for the Cahn{Hilliard equation with a concentration dependent diiusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values 1 and it is shown that the solution is bounded by 1 in magnitude. Finally applications of our method to other degenerate fourth order parabolic equations are discussed.

In this paper, using theory of attractors for multi-valued semiflows and semiprocesses, we prove the existence of compact attractor for a semilinear degenerate parabolic equation involving the Grushin operator in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. Mathematics Subject Classification: 35B41, 35K65, 35D05