Related papers: Reduction operators of variable coefficient semili…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued…
Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto…
The classical Ka\v{c}anov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation,…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
In this paper, we consider the semilinear wave equation involving the nonlinear damping term $g(u_t) $ and nonlinearity $f(u)$. The well-posedness of the weak solution satisfying some additional regularity is achieved under the wider ranges…
We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form $$\partial_tu-A\varphi(u)=0.$$ These equations are possibly degenerate nonlinear diffusion…
This paper is concerned with the resolution of an inverse problem related to the recovery of a scalar (potential) function $V$ from the source to solution map, of the semi-linear equation $(\Box_{g}+V)u+u^3=0$ on a globally hyperbolic…
In this paper, we describe the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We…
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special…
We study the symmetry reduction of nonlinear evolution and wave type differential equations by using operators of non-point symmetry. In our approach we use both operators of classical and conditional symmetry. It appears that the…
In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large $x$ and/or $t$) invariant under a group $G$ which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential…
Variational regularization and the quasisolutions method are justified for unbounded closed, possibly nonlinear, operators. The argument is quite simple and yields general results.
In this paper, we investigate the existence and multiplicity of weak solutions to problems involving a superposition operator of the type $$\int_{[0, 1]}(- \Delta)^s u d \mu(s),$$ for a signed measure $\mu$ on the interval of fractional…
Weakly nonlinear amplitude equations are derived for the onset of spatially extended patterns on a general class of n-component bulk-surface reaction-diffusion systems in a ball, under the assumption of linear kinetics in the bulk and…
A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new…
We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega, \] where $0 < q < 1$,…
We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $\alpha$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional…
In the paper, we discuss the reconstruction of scalar parameters in a linear diffusion equation with fractional in time differential operators and with additional nonlocal (convolution) terms, which incorporate memory effects in models.…
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown,…