WELL- POSEDNESS OF THE ROTHE DIFFERENCE SCHEME FOR REVERSE PARABOLIC EQUATIONS

well- posedness of the rothe difference scheme for reverse parabolic equations

Well-posedness of parabolic differential and difference equations

The stable difference scheme for the approximate solution of the initial value problem ( ) ( ) ( ) ( ) 1 2 , t du t D u t Au t f t dt + + = ( ) 0 1, 0 0 t u < < = for the differential equation in a Banach space E with the strongly positive operator A and fractional operator 1 2 t D is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is...

Well-Posedness of Parabolic Differential and Difference Equations with the Fractional Differential Operator

The stable difference scheme for the approximate solution of the initial value problem du(t) dt + D 1 2 t u(t) + Au(t) = f(t), 0 < t < 1, u(0) = 0 for the differential equation in a Banach space E with the strongly positive operator A and fractional operator D 1 2 t is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In...

Well-posedness for Non-isotropic Degenerate Parabolic-hyperbolic Equations

We develop a well-posedness theory for solutions in L to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; a...

Well-posedness Results for Triply Nonlinear Degenerate Parabolic Equations

We study the well-posedness of triply nonlinear degenerate ellipticparabolic-hyperbolic problem b(u)t − div ã(u, ∇φ(u)) + ψ(u) = f, u|t=0 = u0 in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b, φ and ψ are supposed to be continuous non-decreasing, and the nonlinearity ã falls within the Leray-Lions framework. Some restrictions are imposed on the dependence...

Well-posedness of parabolic equations containing hysteresis with diffusive thresholds

In the paper, we develop a theory of reaction-diffusion equations containing discontinuous hysteresis operator — the so-called non-ideal relay. The nonideal relay (or a bi-stable relay, or lazy switch) is the most basic, yet nontrivial hysteresis operator. The state (output) of the non-ideal relay switches from −1 to 1 when the input exceeds a threshold value x ∈ R and switches back to state −1...