Nonlinear Biharmonic Equation Research Articles

Positive Radial Symmetric Solutions of Nonlinear Biharmonic Equations in an Annulus

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵2u=f(u,▵u) on an annular domain Ω in RN with the Navier boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:R+×R−→R+ is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f(ξ,η) to have superlinear or sublinear growth on ξ,η as |(ξ,η)|→0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones.

Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs

In this paper, let G=(ZN,E,μ,ω) be a weighted lattice graph, where μ is a uniformly bounded measure and ω is a positive symmetric weight. We are concerned with the following biharmonic Choquard equation with a nonlocal nonlinearity on ZN, for any α∈(0,N),Δ2u−Δu+hu=(∑y∈ZN,y≠xF(u(y))d(x,y)N−α)f(u), where h denotes a function on ZN, f:R→R is a continuous function, F(u)=∫0uf(s)ds represents the primitive function of f, and d(x,y) stands for the distance between x and y. In particular, under some suitable conditions on h, we prove that if the nonlinearity f satisfies distinct growth conditions, then there exists a mountain-pass solution and a ground state solution respectively. As far as we know, there are no existing results for the biharmonic Choquard equation with the nonlinearity f on weighted lattice graphs.

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>

Gradient auxiliary physics-informed neural network for nonlinear biharmonic equation

Physics-informed neural network (PINN) has gained wide attention for solving forward and inverse problems of partial differential equation (PDE) from both data-driven and model-driven perspectives. In a short time, various machine learning methods based on PINN have been developed for solving a broad range of PDE. To improve the training accuracy of PINN, the popular approach is to introduce penalty parameters to correct the imbalance among different parts of loss function during model training. However, this approach generally fails to eliminate the model error generated by the boundary condition. To eliminate this issue, we propose a gradient auxiliary physics-informed neural network (GA-PINN) for nonlinear biharmonic equation. The key idea of GA-PINN is to split original biharmonic equation subjected to clamped or simply supported boundary conditions into third order or second order differential equations by introducing gradient auxiliary functions. As a consequence, we can rewrite clamped or simply supported boundary conditions as Dirichlet boundary conditions, then conveniently construct the neural network composite functions to satisfy those Dirichlet boundary conditions. We introduce the dynamic weight method to automatically balance the contributions of different loss terms during training. The capabilities of the proposed GA-PINN by solving several nonlinear biharmonic problems are demonstrated.

Multiple Solutions of a Nonlinear Biharmonic Equation on Graphs

In this paper, we consider a biharmonic equation with respect to the Dirichlet problem on a domain of a locally finite graph. Using the variation method, we prove that the equation has two distinct solutions under certain conditions.

Regularization of a final value problem for a linear and nonlinear biharmonic equation with observed data in $ L^{q} $ space

<abstract><p>In this work, we focus on the final value problem of an inverse problem for both linear and nonlinear biharmonic equations. The aim of this study is to provide a regularized method for the bi-harmonic equation, once the observed data are obtained at a terminal time in $ L^{q}(\Omega) $. We obtain an approximated solution using the Fourier series truncation method and the terminal input data in $ L^{q}(\Omega) $ for $ q \ne 2 $. In comparision with previous studies, the most highlight of this study is the error between the exact and regularized solutions to be estimated in $ L^{q}(\Omega) $; wherein an embedding between $ L^{q}(\Omega) $ and Hilbert scale spaces $ \mathcal{H}^{\rho}(\Omega) $ is applied.</p></abstract>

Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials

We establish the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey–Lorentz spaces. As a by-product we derive estimates on the restriction operator in block spaces. Direct application to solvability of a fourth order nonlinear equation related to higher order boundary conformally invariant problem is considered. By employing a nonvariational approach, we obtain a unique solution under suitable smallness conditions on the boundary data and potentials prescribed in Morrey–Lorentz spaces which allow for singular functions. Moreover, these solutions are shown to be C∞ in the interior, Cloc2,μ(R+n+1¯), μ∈(0,1) in some special case and satisfy interesting qualitative properties including positivity. The results are extended to a larger class of problems involving the polyharmonic operator. In particular, the higher order nonlocal equation (−Δ)2m−12v=K(x)|v|σ−1v+H(x)v+ginRn,m∈Nfor 2≤2m<n+1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ>n/(n−2m+1). We also prove that no positive solution in the latter framework exists if σ≤n/(n−2m+1).

Local behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation Δ2u=upinB2∖{0} with an isolated singularity, where the punctured ball B2∖{0}⊂Rn with n>5 and p>(n+4)/(n−4). We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for −Δu.

On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

Multiple solutions for generalized biharmonic equations with singular potential and two parameters

We investigate a more general nonlinear biharmonic equation Δ 2 u − β Δ p u + V λ ( x ) u = f ( x , u ) in ℝ N , where Δ2:=Δ(Δ) is the biharmonic operator, N≥1, λ>0 and β∈ℝ are parameters, Δpu= div(|∇u|p−2∇u) with p≥2. Differently from previous works on biharmonic problems, we replace Laplacian with p-Laplacian, and suppose that V(x)=λa(x)−b(x) with λ>0 and b(x) can be singular at the origin, in particular we allow β to be a real number. Under suitable conditions on Vλ(x) and f(x,u), the multiplicity of solutions is obtained for λ>0 sufficiently large. Our analysis is based on variational methods as well as the Gagliardo–Nirenberg inequality.

Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $$\begin{aligned} \Delta ^2 u = u^p\quad \text{ in }\; B_1\backslash \{0\} \end{aligned}$$with an isolated singularity, where the punctured ball \(B_1 \backslash \{0\} \subset {\mathbb {R}}^n\) with \(n\ge 5\) and \(\frac{n}{n-4}< p < \frac{n+4}{n-4}\). This equation is relevant for the Q-curvature problem in conformal geometry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for \(-\Delta u\). We also give a new method to prove removable singularity theorem for nonlinear higher order equations.

An approximate solution for a nonlinear biharmonic equation with discrete random data

In this paper, we study a problem of finding the solution for the nonlinear biharmonic equation Δ2u=f(x,t,u(x,t)) from the final data. By using a simple example, the ill-posedness of the present problem with random noise is demonstrated. The Fourier method is conducted in order to establish an estimator for the mild solution (called regularized solution) and the convergence results in some different cases are proposed. Finally, numerical experiments are presented for showing that this regularization method is flexible and stable.

Non-linear bi-harmonic Choquard equations

This note studies the fourth-order Choquard equation \begin{document}$ i\dot u+\Delta^2 u\pm (I_\alpha *|u|^p)|u|^{p-2}u = 0 . $\end{document} In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.

Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling

In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G=(V,E), which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equationΔ2u−Δu+(λa+1)u=|u|p−2u on G=(V,E). Under some suitable assumptions, we prove that for any λ>1 and p>2, the equation admits a ground state solution uλ. Moreover, we prove that as λ→+∞, the solutions uλ converge to a solution of the equation{Δ2u−Δu+u=|u|p−2u,inΩ,u=0,on∂Ω, where Ω={x∈V:a(x)=0} is the potential well and ∂Ω denotes the boundary of Ω.

Regularization of a final value problem for a nonlinear biharmonic equation

In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in uniformly with respect to the space coordinate under some a priori assumptions on the solution.

Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent

For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation \[ \Delta^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable singularity at the origin. We show that $|x|^{\frac{n-4}{2}} u$ is a periodic function of $\ln |x|$ and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior near singularities and for the $Q$-curvature problem in conformal geometry.

Existence of positive ground solutions for biharmonic equations via Pohožaev-Nehari manifold

We investigate the following nonlinear biharmonic equations with pure power nonlinearities: \begin{equation*} \begin{cases} \triangle^2u-\triangle u+V(x)u= u^{p-1}u &\text{in } \mathbb{R}^N, u> 0 &\text{for } u\in H^2(\mathbb{R}^N), \end{cases} \end{equation*} where $2< p< 2^*={2N}/({N-4})$. Under some suitable assumptions on $V(x)$, we obtain the existence of ground state solutions. The proof relies on the Pohožaev-Nehari manifold, the monotonic trick and the global compactness lemma, which is possibly different to other papers on this problem. Some recent results are extended.

Existence and concentration of a nonlinear biharmonic equation with sign-changing potentials and indefinite nonlinearity

We consider the following nonlinear biharmonic equations: \t\t\tΔ2u−Δu+Vλ(x)u=f(x,u),in RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta^{2} u-\\Delta u+ V_{\\lambda }(x)u=f(x,u),\\quad \\text{in } \\mathbb{R}^{N}, $$\\end{document} where V_{lambda }(x) is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.