Reaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

We obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu + V(x)u–au = f where a ≥ 0. The conditions are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

When two miscible fluids, such as glycerol (glycerin) and water,are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients existduring some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical modelconsisting of the diffusion equation with convective terms and ofthe Navier-Stokes equations with the Korteweg stress.We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain.We study the longtime behavior of the solution and show that it convergesto the uniform composition distribution with zero velocity field.We also present numerical simulations of miscible drops and show howtransient interfacial phenomena can change their shape.

The paper is devoted to analysis of an elliptic-algebraic system ofequations describing heat explosion in a two phase medium filling a star-shaped domain. Three typesof solutions are found: classical, critical andmultivalued. Regularity of solutions is studied as well as theirbehavior depending on the size of the domain and on the coefficient ofheat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.