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We investigate a stochastic partial differential equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space-time white noise. We introduce a notion of weak…
Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems. The present paper focuses…
We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational…
We study the growth of a periodic pattern in one dimension for a model of spinodal decomposition, the Cahn-Hilliard equation. We particularly focus on the intermediate region, where the non-linearity cannot be negected anymore, and before…
We study irreversible evolutionary processes with a general energetic notion of stability. We dedicate this contribution to releasing three nonlinear variational solvers as modular components (based on FEniCSx/dolfinx) that address three…
We address the system of partial differential equations modeling motion of an elastic body interacting with an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by a…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of…
We consider semilinear equations of the form p(D)u=F(u), with a locally bounded nonlinearity F(u), and a linear part p(D) given by a Fourier multiplier. The multiplier p(\xi) is the sum of positively homogeneous terms, with at least one of…
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping…
In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Using the the Faedo-Galerkin method we establish the global existence of the solution of the…
In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a…
For a class of evolution equations that possibly have only local solutions, we introduce a stochastic component that ensures that the solutions of the corresponding stochastically perturbed equations are global. The class of partial…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
This paper presents a detailed study on analytical solutions to a general nonlinear boundary-value problem in finite deformation theory. Based on canonical duality theory and the associated pure complementary energy principle in nonlinear…
We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-\Delta)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data…
A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.
First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation…
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential…
Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the…