Some parabolic variational inequalities with variable exponent of nonlinearity: unique solvability and comparison theorems

The aim of this article is to study a variant of the Poincaré inequality in Sobolev spaces $$W^{k,p(\cdot )}(\Omega ) $$ W k , p ( · ) ( Ω ) where $$\Omega $$ Ω is a given open set of $$\mathbb {R}^{n}$$ R n with finite Lebesgue measure and, consequently, the Friedrichs type inequality.

A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.

We study the solution theory of singular linear switched systems with inputs (also known as switched descriptor systems). These systems are highly relevant in many applications; in particular, in economics the well-known dynamic Leontief model with changing coefficient matrices falls into this class. Theorem 5.1 in the paper by Anh et al. (2019) stated that if a singular linear switched system is jointly index-1, then there exists an explicit surrogate switched system having identical solution behavior for all switching signals. However, it was not clear yet whether the jointly index-1 condition is a necessary and sufficient condition for the existence and uniqueness of a solution. Furthermore, it was also not clear what conditions are actually required to guarantee existence and uniqueness of solutions for particular switching signals only. In this article, we provide necessary and sufficient conditions for existence and uniqueness of solutions for singular linear switched systems with respect to fixed switching signals (both mode sequences and switching times are fixed), fixed mode sequences (switching times are arbitrary), and arbitrary switching signals (both mode sequences and switching times are arbitrary). In all three cases we provide an explicit surrogate system with the same solution set; our approach improves the results presented in Anh et al. (2019) as the coefficient matrices describing the transition from x(k) to $$x(k+1)$$ x ( k + 1 ) only depend on original system matrices at time k and $$k+1$$ k + 1 and not on $$k-1$$ k - 1 as in Anh et al. (2019). We illustrate the theoretical findings with the dynamic Leontief model and investigate the solvability properties of discretizations of continuous-time singular systems.

We consider stationary anisotropic higher-order partial integrodifferential equations with variable exponents of nonlinearity. It is assumed that the unknown function may nonlinearly enter into the integrand. We establish the conditions of existence and uniqueness of generalized solutions of the Dirichlet problem for these equations.

This paper discusses the existence and uniqueness of strong solutions for the first order homogeneous linear generalized distributed parameter system and nonhomogeneous linear generalized distributed parameter system by using the generalized operator semigroup theory. And sufficient conditions of existence and uniqueness of strong solutions and constructive expressions of strong solutions are given in the paper. Further, the existence of strong solutions of semilinear generalized distributed parameter system is studied by mild solutions.

In Sobolev spaces with variable exponent, we consider the problem for a semilinear hyperbolic variational inequality of the third order. We establish conditions for the existence of a solution u of this problem such that u ∈ L ∞((0, T); V 1,0(Ω)), u t ∈ L ∞((0, T); V 1,0(Ω)) ∩ L p(x)(Q T ), and u tt ∈ L ∞((0, T); L 2(Ω)), where V 1,0(Ω) ⊂ H 1(Ω).

The initial-boundary value problems for parabolic equations with variable exponents of nonlinearity and time depended delay are considered. Existence and uniqueness of solutions of these problems are proved.

In this article, we consider a nonlinear Sobolev type fractional functional integrodifferential equations in a Banach space along with a nonlocal condition. Sufficient conditions for existence, uniqueness and dependence on initial data of local solutions of considered problem are derived by employing fixed point techniques and theory of classical semigroup. Further, we also render the criteria for existence of global solution. At the end, we provide an application to elaborate the obtained results.

Infinite Coupled Riccati Equations Arising in Certain Optimal Control Problem

The solvability of a delay differential equation arising in the construction of quadratic cost functionals, i.e. Lyapunov functionals, for a linear time-delay system with a constant and a distributed delay is investigated. We present a delay-free auxiliary ordinary differential equation system with algebraically coupled split-boundary conditions, that characterizes the solutions of the delay differential equation and is used for solution synthesis. A spectral property of the time-delay system yields a necessary and sufficient condition for existence and uniqueness of solutions to the auxiliary system, equivalently the delay differential equation. The result is a tractable analytic solution framework to the delay differential equation.

We present a unified approach to study large positive solutions (i.e., u ( x ) → ∞ u(x)\to \infty as x → ∂ Ω x\to \partial \Omega ) of the equation Δ u + h u − k ψ ( u ) = − f \Delta u+hu-k\psi (u)=-f in an arbitrary domain Ω \Omega . We assume ψ ( u ) \psi (u) is convex and grows sufficiently fast as u → ∞ u\to \infty . Equations of this type arise in geometry (Yamabe problem, two dimensional curvature equation) and probability (superdiffusion). We prove that both existence and uniqueness are local properties of points of the boundary ∂ Ω \partial \Omega ; i.e., they depend only on properties of Ω \Omega in arbitrarily small neighborhoods of each boundary point. We also find several new necessary and sufficient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries.

where C and a are some positive constants, is uniquely solvable. Similar classes were constructed for general linear parabolic systems in [2], for semilinear second-order parabolic equations in [3], and for quasilinear higher-order parabolic equations in [4–7]; moreover, the last papers show the dependence of well-posedness classes for mixed problems in unbounded domains on geometric properties of the domains. Later, it turned out that the unique solvability of the Cauchy problem for some semilinear and nonlinear parabolic equations is independent of the behavior of the solution as |x| → +∞ [8–20]. This effect was essentially obtained for the first time in [8] for the spatially one-dimensional equation ut −∆ ( |u|p−2u ) + c|u|q−2u = 0 (1)

In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find u ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and f ∈ F ⁢ ( ⋅ , ⋅ , u ) {f\in F(\cdot,\cdot,u)} such that u ⁢ ( ⋅ , 0 ) = u 0 {u(\cdot,0)=u_{0}} and 〈 u t + A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) , \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; Ψ ⁢ ( u ) = ∫ 0 T ψ ⁢ ( t , u ) ⁢ 𝑑 t {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

In this paper we consider proper orthogonal decomposition (POD) methods that do not include difference quotients (DQs) of snapshots in the data set. The inclusion of DQs have been shown in the literature to be a key element in obtaining error bounds that do not degrade with the number of snapshots. More recently, the inclusion of DQs has allowed to obtain pointwise (as opposed to averaged) error bounds that decay with the same convergence rate (in terms of the POD singular values) as averaged ones. In the present paper, for POD methods not including DQs in their data set, we obtain error bounds that do not degrade with the number of snapshots if the function from where the snapshots are taken has certain degree of smoothness. Moreover, the rate of convergence is as close as that of methods including DQs as the smoothness of the function providing the snapshots allows. We do this by obtaining discrete counterparts of Agmon and interpolation inequalities in Sobolev spaces. Numerical experiments validating these estimates are also presented.

Inequalities in Sobolev Spaces