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Existence and nonexistence of radially symmetric ground states and compact support solutions for a quasilinear equation involving the mean-curvature operator are studied in dependence of the parameters involved. Different tools are used in the proofs, according to the cases considered. Several numerical results are also given: the experiments show a possible lack of uniqueness of the solution and a strong dependence on the space dimension.
In this paper we study elliptic boundary-value problems in bounded non-smooth plane domains and prove a generation result concerning analytic semigroups of linear bounded operators in space of continuous functions. Then we apply such a generation result for bounded non-smooth plane domains to a parabolic integro-differential equation.
The initial-boundary value problems describing motion of a two-dimensional viscoelastic fluid are investigated by using the methods of variational formulation and inequality estimates. Both the exponential and power convergence of the solutions to a steady state solution of the viscoelastic fluid flows are proved under prescribed conditions. The convergences to a stead state solution of the Navier-Stokes flows is a special case of the results.
We are concerned with a system of nonlinear partial differential equations modeling the spread of an epidemic disease through a heterogeneous habitat. Assuming no-flux boundary conditions and $L^1$ data, we prove the existence of at least one weak solution.