We consider the aggregation equation ut = ∇·(∇u−u∇K(u)) in a bounded domain Ω ⊂ R with supplemented the Neumann boundary condition and with a nonnegative, integrable initial datum. Here, K = K(u) is an integral operator. We study the local and global existence of solutions and we derive conditions which lead us to either the stability or instability of constant solutions.

Existence of nontrivial nonnegative equilibrium solutions for age structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive...

In this paper, we investigate the global existence of solutions to a hyperbolic-parabolic model of chemotaxis arising in the theory of reinforced random walks. To get L2-estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and a priori assumptions.

All strictly monotonic solutions of a general functional equation are determined. In a particular case, which plays an essential role in the axiomatization of rank-dependent expected utility, all nonnegative solutions are obtained without any regularity conditions. An unexpected possibility of reduction to convexity makes the present proof possible.

We study nonnegative solutions u(x, t) of the nonlinear parabolic inequalities au ≤ ut −∆u ≤ u in various subsets of R×R, where λ > n+2 n and a ∈ (0, 1) are constants. We show that changing the value of a in the open interval (0, 1) can dramatically affect the blow-up of these solutions.

We obtain Sturmian comparison results for the nonnegative solutions to Dirichlet problems associated with p-Laplacian operators. From Picone-type identities [4, 9], we obtain results comparing solutions of two types of equations. We also present results related to those operators using Piconetype identities.

We show the existence of two nontrivial nonnegative solutions and infinitely many solutions for degenerate p(x)-Laplace equations involving concaveconvex type nonlinearities with two parameters. By investigating the order of concave and convex terms and using a variational method, we determine the existence according to the range of each parameter. Some Caffarelli-Kohn-Nirenberg type problems w...

The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquidsolid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence ...

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states. AM...

Using the nonlinear capacity method, we investigate problem of nonexistence nonnegative monotone solutions for a quasilinear elliptic inequality type Δpu ≥ uq in half-space terms parameters p and q.