Initial-boundary-value problems for anisotropic elliptic-parabolicpseudoparabolic equations with variable exponents of nonlinearity

Abstract

Abstract.Existence and uniqueness of weak solutions of initialboundary value problems for anisotropic elliptic-parabolic-pseudoparabolic equations with variable exponents of nonlinearity are proved. Estimates of the weak solutions of this problems are received. This estimates implies continuous dependence on the input data for the weak solutions of considered problems.2010 MSC. 35K70.

Solutions for higher-order anisotropic elliptic-parabolic equations in time unbounded domains

“…We show that for fixed t 0 and R 0 the left side of inequality (24) converges to zero when m, k → +∞. Let ε > 0 be an arbitrary small number.…”

“…The existence and uniqueness of the function u m is proved in [27] (see also [7] and [24]). We extend u m on Q by zero and this extension is denoted by u m again.…”

Boundary optimal control for systems described by parabolic problem without initial conditions (in Ukrainian)

“…Отриманi рiв-ностi типу рiвностi (16) пiдставимо у (20). Звiдси у результатi простих перетворень отримаємо (18).…”

The Fourier problem for weakly nonlinear integro-differential elliptic-parabolic systems

“…The Fourier problem or, in other words, the problem without initial conditions for systems of evolution equations appears in the modeling of various dynamic processes in nature and economy, when the process is so far removed from the current moment that the initial data practically does not affect the situation at that moment (see, for example, [9]). Such a problem was considered in papers of many mathematicians, in particular, in [4]- [14]. A fairly complete overview of these results can be found in [13].…”

“…The Fourier problem for equations or systems in the case of strong nonlinearity can be uniquely solved without any conditions at infinity (see [4]- [8]). The problem without initial conditions for systems (1) is investigated in [7].…”