Test Function Method Research Articles

Complexiton, complex multiple kink soliton and the rational wave solutions to the generalized (3 + 1)-dimensional kadomtsev-petviashvili equation

Some new exact solutions of the generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation (KPE) are explored in this study. Firstly, the resonant multiple soltion solutions (RMSs) are discussed via employing the linear superposition principle and weight algorithm. Then, by introducing pairs of the conjugate parameters to the RMSs, the complexiton solutions including the non-singular complexiton and singular complexiton solutions are extracted. In addition, the complex multiple kink soliton solutions are also probed by employing the bilinear approach. Finally, we investigate the rational wave solutions via the test function method and symbolic computation. By choosing the appropriate parameters, the graph descriptions of the derived solutions are presented to show the dynamical properties. The outcomes of this work are desirous to bring some new perspective to the study of the complexiton, complex solutions and rational wave solutions to the other PDEs.

Novel interaction solutions to the (3+1)-dimensional Hirota bilinear equation by bilinear neural network method

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the ([Formula: see text])-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields.

Behavior of solutions to semilinear evolution inequalities in an annulus: The critical cases

In the present paper, we consider the parabolic and hyperbolic inequalities with singular potentials and with critical nonlinearities in the annulus domain. The problems are studied with Neumann-type and Dirichlet-type boundary conditions on the boundary. Moreover, we study the systems of problems too. We have proved that the above problems are globally unsolvable in critical cases, thereby filling the gaps the recent results by Jleli and Samet in [J. Math. Anal. Appl. 514: 2 (2022)] and in [Anal. Math. Phys. 12: 90 (2022)]. Proofs are carried out using the method of test functions with logarithmic arguments, which is being developed for the first time in bounded domains.

Non-singular complexiton, singular complexiton and the breather wave solutions to the (2 + 1)-dimensional Sawada-Kotera equation

This work aims to construct some novel exact solutions of the (2 + 1)-dimensional Sawada-Kotera equation (SKE). First, the resonant multiple soliton solutions (RMSSs) are discussed by employing the weight algorithm (WA) and linear superposition principle (LSP). Then, based on the RMSSs, the non-singular complexiton and singular complexiton solutions are developed by taking pairs of the conjugate parameters. Finally, the breather wave solutions (BWSs) of the sin-cosh type are explored through the symbolic computation and test function method. The dynamics of the corresponding extracted solutions are displayed graphically for showing the physical properties. The achievements of this article can provide some new insights to the (2 + 1)-dimensional SKE.

Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

<p>In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.</p>

Well-posedness and blow-up results for a time-space fractional diffusion-wave equation

<abstract><p>In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $ \alpha\rightarrow1^- $. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.</p></abstract>

A NONEXISTENCE RESULT FOR DEGENERATE PARABOLIC EQUATIONS INVOLVING ADVECTION TERMS AND WEIGHTS

In this paper, we are concerned with the following equation vt − Δλv + ∇λw · ∇λv = h(x)vp (x, t) ∈ RN × R. Here, p is a real number, w is a smooth function, h ≥ 0 is a weight function which is continuous function satisfying some growth condition at infinity, Δλ is a sub-elliptic operator which is defined by Δλ = XN i=1 ∂xi (λ2i ∂xi ) and ∇λ is the corresponding gradient operator associated to Δλ. By using a kind of maximum principle and the test function method, we establish the nonexistence of positive supersolutions of the above equation.

Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group

Abstract In this article, the Liouville theorems for the Kirchhoff-type parabolic equations on the Heisenberg group were established. The main technique for proving the result relies on the method of test functions.

Bäcklund transformation and interaction solutions of a generalized Kadomtsev–Petviashvili equation with variable coefficients

Kadomtsev–Petviashvili (KP) equation is a classical nonlinear evolution equation applied in many fields such as plasma physics, fluid dynamics and engineering. In general, variable coefficients are of value in reflecting the inhomogeneities of media, nonuniformities of boundaries, and external forces. This paper mainly studies some exact solutions of a generalized KP equation with variable coefficients by using Hirota bilinear method. Based on the bilinear form, one-, two-, N-soliton solution, Bäcklund transformation and Lax pair are obtained. With the use of the test function method, lump solution and interaction solutions of lump-stripe and lump-soliton types are constructed, and the structure of the solutions are shown in three-dimensional figures. The analysis on the interaction solutions reveals new dynamic behaviors and periodic evolutions.

On the Global Nonexistence of a Solution for Wave Equations with Nonlinear Memory Term

The paper is devoted to the problem of the local existence for a solution to a nonlinear wave equation, with the dissipation given by a nonlinear form with the presence of a nonlinear memory term. Moreover, the global nonexistence of a solution is established using the test function method. We combine the Fourier transform and fractional derivative calculus to achieve our goal.

On the absence of global weak solutions for a nonlinear time-fractional Schrödinger equation

In this paper, an initial value problem for a nonlinear time-fractional Schrödinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions.

Dynamics of complexiton, Y-type soliton and interaction solutions to the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation in liquid with gas bubbles

This paper focuses on some novel exact solutions of the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation (KSE). Based on the linear superposition principle (LSP) and weight algorithm (WA), we construct the complexiton solutions by introducing the pairs of the conjugate parameters. Besides, the Y-type soliton solutions are extracted by imposing the resonant conditions on the N-soliton solutions. Finally, two different types of the interaction solutions (ISs) are also explored by exerting the test function method and symbolic computation. The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. To the best of our knowledge, the attained outcomes of this article are brand new and have never been reported in other literature, and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better.

On the rate of convergence in homogenization of time-fractional Hamilton–Jacobi equations

Here, we consider periodic homogenization for time-fractional Hamilton–Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the incompatibility between the function used in the doubling variable method, and the non-locality of the Caputo derivative. Our approach is to provide a lemma to prove the rate of convergence without the doubling variable method with respect to the time variable, which is a key ingredient.

Darcy’s Law for Porous Media with Multiple Microstructures

In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar’s method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.

On the Absence of Weak Solutions of Nonlinear Nonnegative Higher Order Parabolic Inequalities with a Nonlocal Source

The paper proves the absence of solutions of semilinear parabolic inequalities and higher order systems with a singular potential and nonlocal sources. The proofs are based on the test function method developed by E. Mitidieri and S.I. Pokhozhaev.

Mixed Lump-Stripe Soliton Solutions to a New Extended Jimbo-Miwa Equation

In this paper, the localized properties of lump and interaction solutions to a new extended Jimbo-Miwa (EJM) equation are studied. Based on the Hirota bilinear method and the test function method, the exact solutions of the EJM equation are discussed; the lump soliton solution, lump-kink soliton solution, and periodic lump solution are obtained. Furthermore, the dynamic properties of the obtained solutions are also discussed by graphical simulation. As far as we know, the obtained results have not been reported.

On global existence for semilinear wave equations with space-dependent critical damping

The global existence for semilinear wave equations with space-dependent critical damping $\partial_{t}^{2} u - \Delta u + \frac{V_0}{|x|} \partial_{t} u = f(u)$ in an exterior domain is dealt with, where $f(u) = |u|^{p-1} u$ and $f(u) = |u|^{p}$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata–Todorova–Yordanov [J. Math. Soc. Japan (2013), 183–236] but the argument in this paper clarifies the precise dependence of the location of the support of initial data. The blowup phenomena are verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.

Nonexistence results of solutions of parabolic-type equations and systems on the Heisenberg group

In this paper, we present necessary conditions for the existence of weak solution of the parabolic-type equations and systems on the Heisenberg group. The main technique for proving the results relies on the method of test functions.

Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term

Abstract This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

Blow-Up Solutions for the Space-Time Fractional Evolution Equation

This paper focuses on the blow-up solutions of the space-time fractional equations with Riemann–Liouville type nonlinearity in arbitrary-dimensional space. Using the Banach mapping principle and the test function method, we establish the local well-posedness and overcome the difficulties caused by the fractional operators to obtain the blow-up results. Furthermore, we get the precise lifespan of blow-up solutions under special initial conditions.