Existence of global solutions to reaction-diffusion systems with nonhomogeneous boundary conditions via a Lyapunov functional.
Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in .
Existence of periodic solutions for a class of nonlinear evolution equations.
Existence of periodic solutions for semilinear parabolic equations
In this paper, we are concerned with the semilinear parabolic equation ∂u/∂t - Δu = g(t,x,u) if u = 0 if , where is a bounded domain with smooth boundary ∂Ω and is T-periodic with respect to the first variable. The existence and the multiplicity of T-periodic solutions for this problem are shown when g(t,x,ξ)/ξ lies between two higher eigenvalues of - Δ in Ω with the Dirichlet boundary condition as ξ → ±∞.
Existence of positive solutions for some nonlinear parabolic equations in the half space.
Existence of pseudo almost automorphic solutions for the heat equation with -pseudo almost automorphic coefficients.
Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities
We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in and b(x,u₀) ∈ L¹(Ω).
Existence of solution a of nonlinear degenerate systems of parabolic variational inequalities
Existence of solution to transpiration control problem.
Existence of solutions and monotone iterative method for infinite systems of parabolic differential-functional equations
We consider the Fourier first boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations. To prove the existence and uniqueness of solution, we apply a monotone iterative method using J. Szarski's results on differential-functional inequalities and a comparison theorem for infinite systems.
Existence of solutions for a degenerate nonlinear evolution equation.
We consider a model of generalized Cahn-Hilliard equations with a logarithmic free energy and a degenerate mobility, and obtain a result on the existence of solutions.
Existence of solutions for a degenerate seawater intrusion problem.
Existence of solutions for a model of self-gravitating particles with external potential
We study the existence of solutions to a nonlinear parabolic equation describing the temporal evolution of a cloud of self-gravitating particles with a given external potential. The initial data are in spaces of (generalized) pseudomeasures. We prove existence of local and global-in-time solutions, and also a kind of stability of global solutions.
Existence of solutions for a nonlinear discrete system involving the -Laplacian
The existence of solutions for boundary value problems for a nonlinear discrete system involving the -Laplacian is investigated. The approach is based on critical point theory.
Existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium
existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium
The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.
Existence of solutions for degenerate sweeping processes.
Existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions and its applications
We establish the existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions. The energies associated to these evolution equations are quadratic forms. Our approach is based on application of the Schaefer fixed-point theorem combined with the continuity method.
Existence of solutions for infinite systems of parabolic equations with functional dependence
The Cauchy problem for an infinite system of parabolic type equations is studied. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. We prove the existence and uniqueness of a bounded solution under Carathéodory type conditions and its differentiability, as well as the existence and uniqueness in the class of functions satisfying a natural growth condition. Both results are obtained by the fixed point method.
Existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces with anti-periodic boundary conditions
The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet -Laplace operator.