We study a parabolic ODE system modeling tumour growth proposed by Othmer and Stevens [8]. In use of the transformation of Levine and Sleeman [6], we reduce it to a hyperbolic equation with strong dissipation. Then, we show the existence of collapse in arbitrary space dimension by the method of energy.

We consider systems of nonlinear parabolic stochastic partial differential equations with dynamical boundary conditions. These boundary conditions are qualitatively different from the standard, like Dirichlet, or Neumann, or Robin boundary conditions. Such conditions contain a time derivative and can be used to describe mathematical models with a dynamics on the boundary. In our model the noise is acting in the domain but also on the boundary and is presented as the temporal generalized derivative of an infinite-dimensional Wiener processes. In addition, we have coefficients for the spatial differential operators depending on space and time. We prove existence and uniqueness of mild solutions to these stochastic partial differential equations and study the properties of these solutions. The stochastic equation on the boundary contains a parameter. If this parameter becomes small, then the equation on the boundary has an interpretation in a fast time scale. We prove the relative compactness of the distribution of the solution if the parameter mentioned tends to 0.

We study the global existence and long-time behavior of the solutions to a special reaction-diffusion system arising in mathematical population dynamics, with kinetics occurring on distinct spatial domains. First, we give a comprehensive description of the dynamics of the solutions of the underlying system of ordinary differential equations. Next, we analyze a simpler problem where the spatial domain is the same for all the partial differential equations. Last, we prove global existence for the original problem; we offer a conjecture concerning the large-time behavior of solutions and give some hints on its derivation and proof.

We estimate the blow-up rate and then improve some existing uniqueness results for boundary blow-up solutions to certain quasilinear elliptic equations with a weight function. The weight function is allowed to vanish on the part of the boundary where the solution blows up. Our approach is based on the construction of certain upper and lower solutions on small annuli with partial boundary blow-up, and on a modified version of an iteration technique due to Safonov.

By means of variational method, we study a singular critical growth semilinear elliptic problem: $-\Delta{u}=Q(x)|u|^{2^*-2}{u}+\mu \frac{u}{|x|^2}+\lambda u,$ $u\in H^1_0(\Omega)$, where $2^*=\frac{2N}{N-2},$ $N\geq 7,$ $0 <\mu <\frac{(N-2)^2}{4},$ $\lambda>0$, and $Q(x)$ is a positive function on $\overline{\Omega}$. By investigating the effect of the coefficient of the critical nonlinearity, we prove the existence of sign-changing solutions.

We study the blowup behavior of an equation arising from plane curves expansion. We give a precise description of the blowup rate up to the addition of a constant. We also investigate other related properties of this equation.

The purpose of the present paper is to continue the program of study of elliptic boundary-value problems on Lipschitz domains with boundary data in quasi-Banach Besov spaces $B_s^{p,p}(\partial \Omega)$, initiated in [13]. Introducing a modified square function which is well-adapted for handling data with a fractional amount of smoothness, we establish the well-posedness of the Dirichlet and Neumann boundary problems for the Laplacian in Lipschitz domains, for a range of indices which includes values of $p$ less than $1$. An important ingredient in this regard is establishing suitable square-function estimates for a singular integral of potential type.

We consider an initial-boundary-value problem for a nonlinear differential system consisting of one equation of parabolic type coupled with an $n \times n$ semilinear hyperbolic system of first order. This system of equations describes the compressible, $ ( n +1 )$-component, miscible displacement in a porous medium, without including effects of molecular diffusion and dispersion. Assuming some regularity conditions on the data, we prove the existence (locally in time) and uniqueness of classical solutions.

We give sufficient conditions for the existence of almost-periodic solutions of the second-order differential system $u'' = f(t,u,u')$, which satisfies the maximum principle. Our hypotheses do not impose the uniqueness of bounded solutions.

We study the existence of bifurcation points for variational inequalities with strongly indefinite quadratic part. We use a nonsmooth variational framework. An application to an elliptic system with an obstacle is given.