We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand side depends on the derivative of unknown function with deviating argument.
We consider the Fourier first initial-boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations of parabolic type. The right-hand sides of the system are functionals of unknown functions. The existence and uniqueness of the solution are proved by the Banach fixed point theorem.
We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here and the solution belongs to the so-called grand Sobolev space . This is the proper space when the right hand side is assumed to be only -integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.
L’existence de solutions holomorphes locales d’équations aux dérivées partielles d’ordre infini à coefficients holomorphes de type spécial est étudiée.
We study a general class of nonlinear elliptic problems associated with the differential inclusion in Ω where . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general -data.
The paper deals with the initial boundary value problem of Robin type for parabolic functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in the classical sense. A theorem on the existence of a classical solution is proved. Our formulation and results cover differential equations with deviated variables and differential integral problems.
We consider a nonlinear evolution inclusion defined in the abstract framework of an evolution triple of spaces and we look for extremal periodic solutions. The nonlinear operator is only pseudomonotone coercive. Our approach is based on techniques of multivalued analysis and on the theory of operators of monotone-type. An example of a parabolic distributed parameter system is also presented.
We prove the local existence of solutions for equations of motion of a viscous compressible barotropic fluid in a domain bounded by a free surface. The solutions are shown to exist in exactly those function spaces where global solutions were found in our previous papers [14, 15].
Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where , , G is a bounded domain with (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real function. For the equation (1) we consider the Dirichlet problem with the boundary condition (2) u(x) = h(x) for x∈ ∂G. We use Chaplygin’s method [5] to prove that problem (1), (2) has at least one regular solution in a suitable class of functions. Using the method of upper and lower...