We investigate the global time existence of smooth solutions for the ShigesadaKawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global W 1,p -estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.

Global weighted L p -estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a weight in a Muckenhoupt class. Under a smallness condition on the mean oscillation of the coefficients with the weight and a Reifenberg flatness condition on the boundary of the domain, we establish a weighted gradient estimate for weak solutions of the equation. A class of degenerate coefficients satisfying the smallness condition is characterized. A counter example to demonstrate the necessity of the smallness condition on the coefficients is given. Our W 1,p -regularity estimates can be viewed as the Sobolev's counterpart of the Hölder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni in 1982.

ABSTRACT. The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids and study the initial boundary value problem for the resulting degenerate parabolic equation for pressure with the time-dependent flux boundary condition. We estimate L ∞ -norm for pressure and its time derivative, as well as other Lebesgue norms for its gradient and second spatial derivatives. The asymptotic estimates as time tends to infinity are emphasized. We then show that the solution (in interior L ∞ -norms) and its gradient (in interior L 2−δ -norms) depend continuously on the initial and boundary data, and coefficients of the Forchheimer polynomials. These are proved for both finite time intervals and time infinity. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are combined with uniform Gronwall-type estimates, specific monotonicity properties, suitable parabolic Sobolev embeddings and a new fast geometric convergence result.

We study quasilinear elliptic equations of the form div A(x, u, ∇u) = div F in bounded domains in R n , n ≥ 1. The vector field A is allowed to be discontinuous in x, Lipschitz continuous in u and its growth in the gradient variable is like the p-Laplace operator with 1 < p < ∞. We establish interior W 1,q -estimates for locally bounded weak solutions to the equations for every q > p, and we show that similar results also hold true in the setting of Orlicz spaces. Our regularity estimates extend results which are only known for the case A is independent of u and they complement the wellknown interior C 1,α -estimates obtained by DiBenedetto [9] and Tolksdorf [33] for general quasilinear elliptic equations. 1 2 A(x)ξ with the matrix A(x) being uniformly elliptic and bounded, Kinnunen and Zhou [20] obtained interior W 1,q -estimates when A(x) ∈ V MO, i.e. A(x) is of vanishing mean oscillation. Recently, Byun and Wang [2] (see also [3]) were able to obtain W 1,q -estimates for (1.5) under the assumption that the BMO modulus of A in the x variable is sufficiently small. Our obtained estimates in Theorem 1.1 are the same spirit as [2] but for general quasilinear elliptic equations of the form (1.1).The proofs of W 1,q -estimates for solutions to (1.5) in the above mentioned work use the perturbation technique from [4-6] and rely essentially on the central fact that equations of this type are invariant with respect to dilations and rescaling of domains. Unfortunately, this is no longer true for equations of the general form (1.1) and this presents a serious obstacle in deriving W 1,q -estimates for their solutions. Our idea to handle this issue is to enlarge the class of equations under consideration in a suitable way by considering the associated quasilinear elliptic equations with two parameters (see equation (2.3)). The class of these equations is the smallest one that is invariant with respect to dilations and rescaling of domains and that contains equations of the form (1.1). Given the invariant structure, a key step in our derivation of W 1,q -estimates for the solution u is to be able to approximate ∇u by a good gradient in L p norm in a suitable sense (see Corollary 5.2). B 3 A(x, u, ∇u), ∇u − ∇v dx + B 3 F, ∇u − ∇v dx. This gives J := B 3 a(x, v, ∇u) − a(x, v, ∇v), ∇u − ∇v dx = B 3 a(x, v, ∇u) − A(x, u, ∇u), ∇u − ∇v dx + B 3 F, ∇u − ∇v dx.

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