Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel system (KS) of parabolic–parabolic type. The global existence of weak solutions to (KS) is established when $q<m+\frac{2}{N}$ (m denotes the intensity of diffusion and q denotes the nonlinearity) without restriction on the size of initial data; note that $q=m+\frac{2}{N}$ corresponds to generalized Fujita?s exponent. The result improves both Sugiyama (2007) [14, Theorem 1] and Sugiyama and Kunii (2006) [15, Theorem 1] in which it is assumed that $q?m$.     

On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems

Summary The Hölder continuity of bounded weak solutions of quasilinear parabolic systems with main part in diagonal form is proved via a parabolic hole-filling technique.This research was supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft     

Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent

In this paper, we prove the Hölder continuity of solutions of parabolic equations containing the p(x, t)-Laplacian. The degree p must satisfy the so-called logarithmic condition.     

Convergence of fundamental solutions of linear parabolic equations under Cheeger–Gromov convergence

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger–Gromov convergence of the underlying manifolds and the uniform L ¹-bound of the solutions. We also prove a local integral estimate of fundamental solutions.     

Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to (see Theorem 3).     

Regularity of solutions of the model Venttsel’ problem for quasilinear parabolic systems with nonsmooth in time principal matrices

The Venttsel’ problem in the model statement for quasilinear parabolic systems of equations with nondiagonal principal matrices is considered. It is only assumed that the principal matrices and the boundary condition are bounded with respect to the time variable. The partial smoothness of the weak solutions (Hölder continuity on a set of full measure up to the surface on which the Venttsel’ condition is defined) is proved. The proof uses the A(t)-caloric approximation method, which was also used in [1] to investigate the regularity of the solution to the corresponding linear problem.     

Periodic solutions of “predator–prey” systems with continuous delay and periodic coefficients

We prove the existence of positive ω-periodic solutions for some “predator–prey” systems with continuous delay of the argument for the case where the parameters of these systems are specified by ω-periodic continuous positive functions.     

On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit L _p -viscosity solutions, which are in C ^1+α for an ${\alpha \in (0, 1)}$ . The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.     

Existence of weak solutions to the Keller–Segel chemotaxis system with additional cross-diffusion

In this paper, we consider the Keller–Segel chemotaxis system with additional cross-diffusion term in the second equation. This system is consisting of a fully nonlinear reaction–diffusion equations with additional cross-diffusion. We establish the existence of weak solutions to the considered system by using Schauder’s fixed point theorem, a priori energy estimates and the compactness results.     

Stability of solutions to linear impulsive systems of Itô differential equations with aftereffect with respect to initial data in part of variables

The method of model equations is used to study the moment stability of solutions to linear impulsive systems of Itô differential equations with aftereffect with respect to initial data in part of variables. Sufficient conditions of stability are obtained in terms of parameters of these systems.     

Global existence of solutions to a parabolic–elliptic chemotaxis system with critical degenerate diffusion

This paper is devoted to the analysis of nonnegative solutions for a degenerate parabolic–elliptic Patlak–Keller–Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, thereby completing previous results on finite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.     

New exact kink solutions and periodic form solutions for a generalized Zakharov–Kuznetsov equation with variable coefficients

In this short letter, with the aid of symbolic computational system Mathematic, new exact solutions including kink solutions, soliton-like solutions and periodic form solutions for a generalized Zakharov–Kuznetsov equation with variable coefficients are obtained using the generalized Riccati equation mapping method.