Solvability of a semilinear generalized distributed parameter system

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.

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 consider parabolic variational inequalities in Sobolev spaces and generalized Lebesgue spaces. Sufficient conditions of existence and uniqueness of solution of these inequalities have been established. We have also proved some comparison theorems.

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.

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.

In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space $\Phi$. Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where $\Phi$ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as $\Phi$-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation, a stochastic integration by parts formula and a stochastic Fubini theorem. Our theory is then applied to provide sufficient and necessary conditions for existence and uniqueness of solutions to linear stochastic evolution equations driven by semimartingale noise taking values in the strong dual $\Phi'$ of $\Phi$. In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.

In this work we study a system of parabolic reaction-diffusion equations which are coupled not only through the reaction terms but also by way of nonlocal diffusivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we find the conditions under which the existence of strong solutions is assured. We establish several blow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.

Global existence and time behaviour of strong solution to a compressible non-Newtonian fluid

In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo −1 T (R N) for T > 0 and the initial density ρ 0 is in L ∞ (R N) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, ·) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 − 1, ρ 0 u 0) ∈ (B N 2 −1 2,∞ (R N) ∩ B N 2 2,∞ (R N)∩L ∞ (R N))×(B N 2 −1 2,∞ (R N)) N. This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.

This article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling assumptions. Then, we report on the results of existence of weak solutions, using the notion of renormalized solutions. Next, existence of strong solutions in the neighborhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted: the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques.

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain $\mathcal{M}=(0,1)_{x}\times(-\pi /2, \pi /2)^d,$ $ d=1,2$ supplemented with suitable boundary conditions. We prove that there exists a solution $u \in \mathcal C ([0, T]; H^1(\mathcal{M})) $ to the initial and boundary value problem for the ZK equation in both dimensions $2$ and $3$ for every $T>0$. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in $3D$. More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures. At the same time, the uniqueness of solutions is still open in $2D$ and $3D$ due to the partially hyperbolic feature of the model.

This paper deals with the local existence and uniqueness of strong solutions for the generalized Boussinesq equations with fractional dissipation. As a corollary, we establish some regularity criteria to guarantee smoothness of solutions.

We revisit the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial density allowing vacuum at infinity, we derive the global existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. In particular, the initial data can be arbitrarily large. This improves our previous work where the initial density has non-vacuum states at infinity. The result could also be viewed as an extension of the study in L{\"u}-Xu-Zhong for the inhomogeneous case to the full inhomogeneous situation. The method is based on delicate spatial weighted estimates and the structural characteristic of the system under consideration. As a byproduct, we get the global existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat conducting Navier-Stokes equations with vacuum at infinity.