A density-dependent chemotaxis-haptotaxis system modeling cancer invasion

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue, initially proposed by Chaplain and Lolas. The model consists of three reaction-diffusion-taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. The equation for cell density includes two bounded nonlinear density-dependent chemotactic and haptotactic sensitivity functions. In the presence of logistic damping, we prove the global existence of a unique classical solution to this model by some delicate a priori estimate techniques.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

The unsaturated flow in porous media with dynamic capillary pressure

In this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two-phase flow in porous media with dynamic capillary pressure-saturation relationship where the relaxation parameter depends on the saturation. Following the approach given in [13] the existence of a weak solution is proved using Galerkin approximation and regularization techniques. A priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test-functions, motivated by the fact that considered PDE allows a natural generalization of the classical Kullback entropy. Finally, a special care was given in obtaining an estimate of the mixed-derivative term by combining the information from the capillary pressure with the obtained a priori estimates on the saturation.     

A semilinear equation with generalized Wentzell boundary condition

In this paper we consider a semilinear equation with a generalized Wentzell boundary condition. We prove the local well-posedness of the problem and derive the conditions of the global existence of the solution and the conditions for finite time blow-up. We also derive an estimate for the blow-up time.     

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.     

Existence for a degenerate Cauchy problem

We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.     

Global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions

For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal L_p-regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level.     

A regularity result for quasilinear parabolic systems

We consider bounded, weak solutions of certain quasilinear parabolic systems of second order. If the solution fulfills a suitable smallness condition, we show that it is H?lder continuous and satisfies an a priori estimate. This is a well known result of Giaquinta and Struwe [3]. Their argument employs the use of Green’s functions, which is completely avoided in our proof. Instead, our crucial tool is a weak Harnack inequality for supersolutions due to Trudinger [7] in connection with a technique developed by L.Caffarelli [1]. Received: 25 September 2006     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

Global existence,nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations

In this paper, we study the Cauchy problem of semilinear heat equations. By introducing a family of potential wells, we first prove the invariance of some sets and isolating solutions. Then we obtain a threshold result for the global existence and nonexistence of solutions. Finally we discuss the asymptotic behavior of the solution.     

Global existence and blowup of a localized problem with free boundary

This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory

We show the existence of monotone in time solutions for a semilinear parabolic equation with memory. The blow-up rate estimate of the solution is known to be a consequence of the monotonicity property.     

Nesting inertial manifolds for reaction and diffusion equations with large diffusivity

We study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations. A priori, we review both local and global dynamics of the system in scales of Banach spaces of Hilbert type and we prove the existence of a universal compact attractor for the equations. Extensions yield the existence of a family of nesting inertial manifolds dependent on the diffusion of the system of equations. It is introduced an upper semicontinuity notion in large diffusivity for inertial manifolds. The limit inertial manifold whose dimension is strictly less than those of the infinite dimensional system of semilinear evolution equations is obtained.     

A p-version finite element method for nonlinear elliptic variational inequalities in 2D

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u _p from the p-version for the obstacle problem. We prove the convergence of u _p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.     

Profile of bubbling solutions to a Liouville system

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in R²${\mathbb{R}}^2$. Then we establish a uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.     

A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids

This paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid. Tumor cells consist of two types of cells: proliferating cells and quiescent cells, which have different chemotactic responses to an extracellular nutrient supply. The model is a free boundary problem for a nonlinear system of reaction-diffusion-advection equations, where the free boundary is the outer boundary of the spheroid. The free boundary condition is quite novel due to different velocity of two types of cells. The global existence and uniqueness of solutions to the model is proved. The proof is based on a fixed point argument, together with the Lp-theory for parabolic equations with the third boundary condition.     

The scalar curvature equation on 2- and 3-spheres

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.Research of first author supported in part by NSF grant 91-03949Research of second author supported by a NSF Postdoctoral Fellowship.Research of third author supported in part by NSF grant 91-02872 and the Ellentuck Fund.