Absence Of Global Solutions Research Articles

Nonexistence results for nonlinear fractional differential inequalities involving weighted fractional derivatives

In this paper, we study the nonexistence of global solutions for certain classes of nonlinear fractional differential inequalities involving weighted fractional derivatives and a singular potential function. Namely, using the test function method with a judicious choice of the test function, we obtain sufficient criteria depending on the parameters of the considered problems, under which we have absence of global solutions. Next, some special cases of the potential function are discussed.

On the absence of global solutions to two-times-fractional differential inequalities involving Hadamard-Caputo and Caputo fractional derivatives

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>

Sufficient Criteria for the Absence of Global Solutions for an Inhomogeneous System of Fractional Differential Equations

A nonlinear inhomogeneous system of fractional differential equations is investigated. Namely, sufficient criteria are obtained so that the considered system has no global solutions. Furthermore, an example is provided to show the effect of the inhomogeneous terms on the blow-up of solutions. Our results are extensions of those obtained by Furati and Kirane (2008) in the homogeneous case.

Absence of Global Solutions for a Fractional in Time and Space Shallow-Water System

An initial boundary value problem for a fractional in time and space shallow-water system involving ψ -Caputo fractional derivatives of different orders is considered. Using the test function method, sufficient criteria for the absence of global in time solutions of the system are obtained.

On absence of global solutions of quasilinear differential-convolutional inequalities

ABSTRACTFrom a unique viewpoint, elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms and nonlocal convolutional terms arising in the description of reaction–diffusion processes, neural networks, and nonlocal phase transitions are investigated. Sufficient conditions of the absence of their global solutions, i.e. necessary conditions of their global solvability, are found.

On the absence of global solutions for some q-difference inequalities

In this paper, we obtain sufficient conditions for the nonexistence of global solutions for some classes of q-difference inequalities. Our approach is based on the weak formulation of the problem, a particular choice of the test function, and some q-integral inequalities.

On the Absence of Global Solutions of a Class of Higher-Order Evolution Inequalities

On the Absence of Global Solutions of a Class of Higher-Order Evolution Inequalities

Absence of Global Solutions of a Mixed Problem for a Schrödinger-Type Nonlinear Evolution Equation

We study the problem of the absence of global solutions of the first mixed problem for one nonlinear evolution equation of Schrodinger type. We prove that global solutions of the studied problem are absent for “sufficiently large” values of the initial data.

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.

On the absence of global solutions to the gauss equation and solutions in external areas

We consider conditions under which the Gauss equation has no solutions defined in the whole space or in areas external with respect to a ball. The absence of solutions in external areas is established in the case when the number of independent variables is more than two. In the two-dimensional case we obtain conditions ensuring the absence of global solutions to the second-order elliptic equation with variable coefficients in its linear part.

The Cauchy-Darboux problem for the one-dimensional wave equation with power nonlinearity

The questions are studied of existence and uniqueness of a global solution to the Cauchy-Darboux problem for the one-dimensional wave equation with power nonlinearity. Under consideration are the existence of local solutions and the absence of global solutions.

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (uxx − u)t + u xxx + uux = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.

Absence of Global Solutions to Systems of Perturbed Parabolic Inequalities with Chipot–Weissler Nonlinearity in the Gradient Term

Abstract For a wide class of systems of nonlinear parabolic inequalities with Chipot-Weissler nonlinearity in the gradient term which depend on two nonnegative parameters λ1 and λ1, we study the nonexistence of global solutions according to the \test function" method. We establish theorems that, when λ1 = λ1 = 0, include and, in some cases, improve, or, in part, yield the results obtained by Es- cobedo and Herrero; Escobedo and Levine; Galaktionov;Levine, Lieberman and Meier; Mitidieri and Pohozaev; Pohozaev and Tesei.

On the absence of global solutions of nonlinear mixed problems

The paper gives examples demonstrating the blow-up phenomenon for solutions of partial differential equations. It is shown that the nature of blow-up of solutions of the mixed problem substantially differs from that of solutions of the Cauchy problem.

Absence of local and global solutions to an elliptic system with time-fractional dynamical boundary conditions

This paper presents extensions of some nonexistence results for elliptic systems with dynamical boundary conditions involving the time-derivatives of integer orders to the case of noninteger order. In particular, we consider a system of Poisson?s equations with time-fractional derivatives of order less than one in the boundary conditions and specify the thresholds of the nonlinearities which lead to the absence of global solutions. The fractional derivatives here are meant in the Riemann-Liouville sense (or in the Caputo sense). We also present necessary conditions for the existence of local solutions.

On the existence or absence of global solutions for the multidimensional version of the second Darboux problem for some nonlinear hyperbolic equations

On the existence or absence of global solutions for the multidimensional version of the second Darboux problem for some nonlinear hyperbolic equations

On the Critical Exponents of Certain Nonlinear Boundary-Value Problems with Biharmonic Operator in the Exterior of a Ball

We establish sufficient conditions for the absence of global solutions of the differential inequality Δ2u≥|u|q in the exterior of a ball. We consider various boundary conditions and show that the critical exponents depend on these conditions. The proofs are based on the test function method developed by Mitidieri and Pokhozhaev.

On the existence or the absence of global solutions of the Cauchy characteristic problem for some nonlinear hyperbolic equations

For wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of the Cauchy characteristic problem in the light cone of the future.

Nonexistence of Global Solutions to a Class of Nonlinear Differential Inequalities and Application to Hyperbolic–Elliptic Problems

posed in Ω × (0, +∞). Here is a an open smooth bounded domain and ϕ is like ϕ(s) = bsγ, γ > 0, a > 0 and ε = ±1. We prove, in certain conditions on f and ϕ that there is absence of global solutions. The method of proof relies on a simple analysis of the ordinary inequality of the type

Absence of Solutions to Higher-Order Evolution Differential Inequalities

Sufficient conditions are given for absence of global solutions to semilinear evolution differential equations and inequalities of higher order (in the time variable). These results generalize the well-known assertions for parabolic and hyperbolic equations. The proofs base on the test function method by Veron, Mitidieri, Pokhozhaev, and Tesei.