Initial-Boundary Value Problems for Anisotropic Parabolic Equations with Variable Exponents of the Nonlinearity in Unbounded Domains with Conditions at Infinity

We deal with the initial-boundary value problems with some restrictions atinfinity for linear and nonlinear anisotropic parabolic second-order equations in unbounded domains with respect to the spatial variables. The weak solutions of our problem in Lebesgue and Sobolev spaces with variable exponents is considered. We prove theorems on the existence and uniqueness of the weak solutions using the method based on Saint- Venant principle, and the monotonicity method. Moreover, we obtain estimate of the weak solutions

Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains

We research the well-posedness of the problem without initial condition for nonlinear parabolic variational inequalities with variable time-delay. To justify our results, we impose some assumptions on the solution behavior and growth of the data-in as time variable tends to −∞. Also, we obtain estimates for weak solutions of this problem

FOURIER PROBLEM FOR WEAKLY NONLINEAR EVOLUTION INCLUSIONS WITH FUNCTIONALS

The Fourier problem or, in other words, the problem without initial conditionsfor evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained.The Fourier problem or, in other words, the problem without initial conditionsfor evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained

Dynamical problems without initial conditions for elliptic-parabolic equations in spatial unbounded domains

We consider a problem without initial conditions for degenerate nonlinear evolution equations with nonlinear dynamical boundary condition in spatial unbounded domains. We obtain sufficient conditions for the well-posedness of this problem without any restrictions at infinity

Almost periodic solutions of anisotropic elliptic-parabolic equations with variable exponents of nonlinearity

We prove the well-posedness of Fourier problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity without any restrictions at infinity. We obtain estimates of the weak solutions and conditions for the existence of periodic and almost periodic solutions. In addition, some properties of the weak solutions of the Fourier problem are considered

Optimal control for systems governed by parabolic equations without initial conditions with controls in the coefficients

We consider an optimal control problem for systems described by a Fourier problem for parabolic equations. We prove the existence of solutions, and obtain necessary conditions of the optimal control in the case of final observation when the control functions occur in the coefficients