Functional separable solutions for two classes of nonlinear equations of mathematical physics

This study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions (specific expressions for these functions are determined by analyzing the arising functional differential equations). The effectiveness of the method is illustrated by examples of nonlinear reaction–diffusion equations and Klein–Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.Comment: See journal reference for teh final published versio

The heat and mass transfer modeling with time delay

Nonlinear hyperbolic reaction-diffusion equations with a delay in time are investigated. All equations considered here contain one arbitrary function. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Exact solutions with a generalized separation of variables are found. For special cases, new exact solutions in the form of a traveling waves are obtained, some of which can be represented in terms of elementary functions. All of these solutions contain free (arbitrary) parameters, so that one can use them to solve modeling problems of heat and mass transfer with relaxation phenomena

Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $\S(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $$\int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t,$$ for all observables $\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t)\}_{t \geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic