Nonlinear Capacity Method Research Articles

A Time-Fractional Differential Inequality of Sobolev Type on an Annulus

Several phenomena from natural sciences can be described by partial differential equations of Sobolev-type. On the other hand, it was shown that in many cases, the use of fractional derivatives provides a more realistic model than the use of standard derivatives. The goal of this paper is to study the nonexistence of weak solutions to a time-fractional differential inequality of Sobolev-type. Namely, we give sufficient conditions for the nonexistence or equivalently necessary conditions for the existence. Our method makes use of the nonlinear capacity method, which consists in making an appropriate choice of test functions in the weak formulation of the problem. This technique has been employed in previous papers for some classes of time-fractional differential inequalities of Sobolev-type posed on the whole space RN. The originality of this work is that the considered problem is posed on an annulus domain, which leads to some difficulties concerning the choice of adequate test functions.

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.

Nonlinear Liouville-type theorems for generalized Baouendi-Grushin operator on Riemannian manifolds

We are concerned with differential inequalities of the form $$ -[d_{M_2}(y_0,y)]^{2\rho_2}\Delta_{M_1} u-[d_{M_1}(x_0,x)]^{2\rho_1}\Delta_{M_2}u\geq V |u|^p\mbox { in } M_1\times M_2, $$ where $M_i$ ($i=1,2$) are complete noncompact Riemannian manifolds, $(x_0,y_0)\in M_1\times M_2$ is fixed, $d_{M_1}(x_0,\cdot)$ is the distance function on $M_1$, $d_{M_2}(y_0,\cdot)$ is the distance function on $M_2$, $\Delta_{M_i}$ is the Laplace-Beltrami operator on $M_i$, $V=V(x,y) > 0$ is a measurable function, and $p>1$. Namely, we establish necessary conditions for existence of nontrivial weak solutions to the considered problem. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifolds $M_i$. Next, we discuss some special cases of potential functions $V$. The proof of our main result is based on the nonlinear capacity method and a result due to Bianchi and Setti (2018) about the construction of cut-off functions with controlled gradient and Laplacian, under certain assumptions on the Ricci curvatures of the manifolds.

Nonexistence results for ultra-parabolic equations and systems involving fractional Laplacian operator in Heisenberg group

In this paper, sufficient conditions are obtained for the nonexistence of solutions to the Cauchy problems for ultra-parabolic equations involving fractional Laplacian operator. The nonexistence result is also extended to the corresponding system. The results to such problems are derived in the Heisenberg group. The proofs of our nonexistence theorems are based on the nonlinear capacity method. This method is based on the choice of a suitable test function in the weak formulation of the considered problems.

On nonexistence of solutions to some time space fractional evolution equations with transformed space argument

Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a [Formula: see text] system of equations with transformed space arguments.

Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group

We first consider a nonlinear time fractional wave equation with a fractional damping posed on the Heisenberg group with Caputo fractional derivatives that interpolate the heat equation and the wave equation with the linear damping. It extends the Caputo–Wisner, the modified Szabo and the fractional Zener models. The non‐linearity accounts for a nonlinear medium. We present the Fujita exponent for blow‐up which sheds light on the admissible nonlinearity in practice. Then, we establish sufficient conditions ensuring non‐existence of local solutions. using the nonlinear capacity method. Furthermore, we extend the analysisto the case of the system to show the flexibility of the method used.

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

In this article, we modify the results obtained by Mitidieri and Pohozaev on sufficient conditions for the absence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of|the Laplace operator and with a nonlinear term of the form a(x)|(mu)|q+ b(x)|u|s. We obtainoptimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of testfunctions. As a result, we prove the absence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction.

Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}} u+\lambda \, \Delta u \ge |u|^{p(x)}, \quad x\in {\mathbb {R}}^N, \end{aligned}$$where \(N\ge 1\), \(\alpha \in (0,2)\), \(\lambda \in {\mathbb {R}}\) is a constant, \(p: {\mathbb {R}}^N\rightarrow (1,\infty )\) is a measurable function, and \((-\Delta )^{\frac{\alpha }{2}}\) is the fractional Laplacian operator of order \(\frac{\alpha }{2}\). A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with variable-exponent nonlinearities. Besides the consideration of variable-exponent nonlinearities, the novelty of this work consists in investigating sign-changing solutions to the considered problems. Namely, to the best of our knowledge, only nonexistence results of positive solutions to fractional elliptic problems were investigated previously. Our approach is based on the nonlinear capacity method combined with a pointwise estimate of the fractional Laplacian of some test functions, which was derived by Fujiwara (Math Methods Appl Sci 41:4955–4966, 2018) [see also Dao and Reissig (A blow-up result for semi-linear structurally damped \(\sigma \)-evolution equations, arXiv:1909.01181v1, 2019)]. Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.

Blowing-up Solutions of Distributed Fractional Differential Systems

We first show that any solution to a nonlinear equation involving a distributed fractional derivative blows-up in a finite time. Then we extend our analysis to a system of nonlinear equations involving distributed fractional derivatives of different orders with different weight functions. Our results rely on the non-linear capacity method.

Blowing-up solutions of the time-fractional dispersive equations

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

Abstract In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

Nonexistence of solutions to Cauchy problems for anisotropic pseudoparabolic equations

We study the nonexistence of nonnegative global weak solutions to the Cauchy problem of anisotropic pseudoparabolic equations and corresponding systems. Using the method of nonlinear capacity, we establish the sufficient conditions for the nonexistence of gobal weak solutions. The result is extended to the case of coupled system of the same type of equations.

On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form i α ∂ t α ω ( t , z ) + a 1 ( t ) Δ ω ( t , z ) + i α a 2 ( t ) ω ( t , z ) = ξ | ω ( t , z ) | p , ( t , z ) ∈ ( 0 , ∞ ) × R N , where N ≥ 1 , ξ ∈ C \ { 0 } and p > 1 , under suitable initial data. To establish our nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefficient functions a 1 , a 2 ∈ L l o c 1 ( [ 0 , ∞ ) , R ) , and provide two illustrative examples.

On critical exponents for weak solutions of the Cauchy problem for a non-linear equation of composite type

We consider the Cauchy problem for a model partial differential equation of third order with non-linearity of the form , where for and . We construct a fundamental solution for the linear part of the equation and use it to obtain analogues of Green’s third formula for elliptic operators, first in a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of the Cauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solution in weighted spaces of bounded and continuous functions. We prove that every solution of the integral equation is a local-in-time weak solution of the Cauchy problem provided that . When , we use Pokhozhaev’s non-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class of initial functions. When , this method enables us to prove that the Cauchy problem has no global-in-time weak solutions for a large class of initial functions.

Blow-up instability in non-linear wave models with distributed parameters

We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space . Using the contraction mapping principle, we prove the local-in-time solubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacity to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up time. For the other, we use a modification of Levine’s method to obtain sufficient conditions for blow-up in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional with the meaning of energy. We also obtain an upper bound for the blow-up time.

Solution blow‐up for a fractional in time acoustic wave equation

We consider the fractional in time acoustic wave equation where 1<α<2, is the Caputo fractional derivative of order α, u=u(t,x), t>0, , is the pressure in the medium, ε is the nonlinear acoustic parameter, ρ0 is the equilibrium density in the medium, and c0 is the equilibrium sound velocity. We study a Cauchy problem for this equation and a mixed boundary value problem in a bounded domain. For each problem, sufficient conditions for the blow‐up of solutions are derived. Moreover, we provide a class of initial data for which there are no classical solutions even locally in time. Our approach is based on the nonlinear capacity method.

Potential Theory for a Nonlinear Equation of the Benjamin–Bona–Mahoney–Burgers Type

For the linear part of a nonlinear equation related to the well-known Benjamin–Bona–Mahoney–Burgers (BBMB) equation, a fundamental solution is constructed, which is combined with the second Green formula to obtain a third Green formula in a bounded domain. Then a third Green formula in the entire space is derived by passage to the limit in some class of functions. The properties of the potentials entering the Green formula in the entire space are examined. The Cauchy problem for a nonlinear BBMB-type equation is considered. It is proved that finding its classical solution is equivalent to solving a nonlinear integral equation derived from the third Green formula. The unique local-in-time solvability of this integral equation is proved by applying the contraction mapping principle. Next, the local-in-time classical solvability of the Cauchy problem is proved using the properties of potentials. Finally, the nonlinear capacity method is used to obtain a global-in-time a priori estimate for classical solutions of the Cauchy problem.

Global Unsolvability of the Burgers Equation with Fractional Time Derivative

We obtain sufficient conditions for the nonexistence of time-global solutions of a fractional analog of the Burgers equation. Examples of solution blowup are considered for the Cauchy problem. The Mitidieri-Pokhozhaev nonlinear capacity method is used.

Nonexistence of global solutions for a time-fractional damped wave equation in a [formula omitted]-times halved space

In this paper, we study the time-fractional damped wave equation CD0αu−Δu+CD0βu=|u|p,t>0,x∈Dk,where 1<α<2, 0<β<1, CD0σ, σ∈{α,β}, is the left-sided Caputo fractional derivative of order σ with respect to the variable time t, p>1, and Dk, k∈{1,2,…,N}, is the k-times halved space given by Dk={x=(x1,x2,…,xN)∈RN:xi>0,i=1,2,…,k}.Using the nonlinear capacity method, we prove that the problem admits no global weak solutions with suitable initial data when 1<p<1+2β(N+k)β+2(1−β).

Nonexistence of travelling wave solution of the Korteweg-de Vries Benjamin Bona Mahony equation

Abstract. This paper is devoted to the Korteweg-de Vries Benjamin Bona Mahony equation in an infinite domain. The paper discusses weak solutions of the Korteweg-de Vries Benjamin Bona Mahony equation without any conditions at infinity. This particular problem arises from the phenomenon of long breaking wave with small amplitude in fluid. In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. For the Korteweg-de Vries Benjamin Bona Mahony equation, we obtain the conditions of blowing-up of travelling wave solutions in finite time. Moreover, there is an explicit upper bound estimate for the wavelength of the corresponding singular traveling wave, depending on the speed of waves. The proof of the results is based on the nonlinear capacity method. In closing, we provide the numerical examples.