Regularity of the solutions for nonlinear biharmonic equations in R^{N}

Regularity of the free boundary in the biharmonic obstacle problem

In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible (NTA) domain, we derive the $C^{1,α}$-regularity of the free boundary in a small ball centered at the origin. From the $C^{1,α}$-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $ C^{3,α}$ up to the free boundary, and therefore $C^{2,1} $. In the end we study an exa…

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $Ω\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $λ\in [0,1]$ we introduce a $λ$-relaxed energy $\H_λ$ for the Hessian energy for maps in $W^{2,2}(Ω,S^4)$ so that each minimizer $u_λ$ of $\H_λ$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_λ$ for $λ\in [0,1)$.

Boundary regularity of stationary biharmonic maps

We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $Ω\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth boundary data, we show that if, in addition, $u$ satisfies a certain boundary monotonicity inequality, then there exists a closed subset $Σ\subset\barΩ$, with $H^{n-4}(Σ)=0$, such that $u\in C^\infty(\barΩ\setminusΣ, N)$.

Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane

This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schrödinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic…

双调和映射热流的正则性与唯一性

Regularity and uniqueness of the heat flow of biharmonic maps

本文在重整化总能量的小性条件下，建立了到单位球面及任意无边界紧致黎曼流形的双调和映射热流的正则性与唯一性，并证明了Hessian能量凸性及时间无穷大处的唯一极限。

Quantitative stratification and higher regularity for biharmonic maps

In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $Ω\subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps …

通过定量分层方法获得双调和映射的最优高阶正则性

Optimal higher regularity for biharmonic maps via quantitative stratification

本文利用定量分层方法改进Breiner-Lamm关于双调和映射的几乎最优正则性结果，并得到极小化双调和映射的最优正则性。

The two obstacle problem for the parabolic biharmonic equation

We consider a two obstacle problem for the parabolic biharmonic equation in a bounded domain. We prove long time existence of solutions via an implicit time discretization scheme, and we investigate the regularity properties of solutions.

Regularity property of some operators on Lipschitz spaces and homogeneous Lipschitz spaces related to biharmonic operator

In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated to the biharmonic operator $Δ^2.$ With this characterizations, we prove the boundedness of the Bessel potentials, fractional integrals, fractional powers, Riesz transforms and multipliers of Laplace transforms type associated to $Δ^2$ on Lipschitz spaces and homogeneous Lipschitz spaces. The proofs of these results need the language of semigroups in an essential way.

Biharmonic wave maps: Local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.

Alt-Caffarelli问题的一个双调和类似问题

A biharmonic analogue of the Alt-Caffarelli problem

该研究针对Dirichlet和Navier边界条件下的Alt-Caffarelli问题双调和类似问题，证明了极小元的存在性、基本性质与$C^{1,α}$正则性，得到Navier问题径向边界数据下的对称性结果，并验证了该正则性的最优性。

On the existence of the biharmonic Green kernels and the the adjoint biharmonic functions

We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.

Biharmonic nonlinear scalar field equations

We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ Δ^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N$$ with a Carathéodory function $g:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$, $N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \int_{\mathbb{R}^N}|u|^2\log |u|\,dx\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|Δu|^2\,dx \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\ma…

A biharmonic equation with singular nonlinearity

We study the biharmonic equation $Δ^2 u =u^{-α}$, $0<α<1$, in a smooth and bounded domain $Ω\subset\RR^n$, $n\geq 2$, subject to Dirichlet boundary conditions. Under some suitable assumptions on $ø$ related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

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.

Singular solutions of the L^2-supercritical biharmonic Nonlinear Schrodinger equation

We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic NLS. These solutions have a quartic-root blowup rate, and collapse with a quasi self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.

Singularly perturbed biharmonic problems with superlinear nonlinearities

We are interested in finding a family of solutions to a singularly perturbed biharmonic equation which has a concentration behavior. The proof is based on variational methods and it is used a weak version of the Ambrosetti-Rabinowitz condition.

双调和非线性薛定谔方程的长时约化

Long-time reduction for the biharmonic nonlinear Schrödinger equation

本文证明在一维高阶色散消失极限下，双调和非线性薛定谔方程在中间正则性区间可长时约化为三次非线性薛定谔方程，并给出指数型逼近界，方法基于正则性保持。

与双调和波映射相关的柯西问题的全局结果

Global results for a Cauchy problem related to biharmonic wave maps

本文在尺度临界 Besov 空间中证明了一类带非泛型二次非线性项的导数双调和波方程对小初值的全局存在性，并由此得到双调和波映射方程在包括球面在内的一类目标流形上的小数据全局正则性。

双调和算子三阶非线性扰动的反问题

Inverse Problems For Third-Order Nonlinear Perturbations Of Biharmonic Operators

该研究针对n≥3维有界域上双调和算子的三阶非线性张量扰动反边界问题，证明边界狄利克雷-诺伊曼映射可唯一确定真非线性三阶张量扰动，证明基于对称张量场的广义动量射线变换反演，指出线性情形的对应问题仍待解决。

Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schrödinger equation

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient $K(x)$ behaves like $\left|x\right|^{-b}$ for $0<b<\min \left\{\frac{N}{2},4\right\} $. We show the local well-posedness in the whole $H^s$-subcritical case, with $0<s\le2$. The difficulties of this problem come from the singularity of $K(x)$ and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces.

非线性双调和算子的一个反问题

An inverse problem for a nonlinear biharmonic operator

本文研究非线性双调和算子的反问题，证明一般非线性项 Q(x,u,∇u,Δu) 可由局部 Cauchy 数据集唯一恢复，方法结合一阶线性化、Runge 逼近与线性化反问题唯一性。

Radial sign-changing solutions to biharmonic nonlinear Schrödinger equations

In this work we obtain three radial solutions of a biharmonic stationary Schrödinger equation, being one positive, one negative and one that changes sign. The Dual Decompostion Method is used to split the natural second order Sobolev space considered in order to apply the appropriate variational approach.

Global solutions for $H^s$-critical nonlinear biharmonic Schrödinger equation

We consider the nonlinear biharmonic Schrödinger equation $$i\partial_tu+(Δ^2+μΔ)u+f(u)=0,\qquad (\text{BNLS})$$ in the critical Sobolev space $H^s(\R^N)$, where $N\ge1$, $μ=0$ or $-1$, $0<s<\min\{\fc N2,8\}$ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^αu$ with $λ\in\mathbb{C},α=\frac{8}{N-2s}$. We prove the existence and uniqueness of the global solutions to (BNLS) for the small initial data.

Stability and intersection properties of solutions to the nonlinear biharmonic equation

We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $Δ^2 ϕ= ϕ^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\geq p_c$. Then, we focus on the supercritical case $p\geq p_c$ and we show that the graphs of no two solutions intersect one another.

四维空间中的双调和非线性向量场方程

Biharmonic nonlinear vector field equations in $\mathbb{R}^4$

本文在四维空间中证明了双调和非线性向量场方程的基态解存在性，并扩展了双调和对数Sobolev不等式。

带组合非线性项的双调和非线性薛定谔方程驻波的存在性与不稳定性

Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities

该研究针对带组合非线性项的双调和薛定谔方程，用新方法证明了归一化基态解的存在性、基态对应限制能量泛函的局部极小值，且驻波会因爆破强不稳定，结果涵盖临界情形。

Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation

We present new singular solutions of the biharmonic nonlinear Schrodinger equation in dimension d and nonlinearity exponent 2σ+1. These solutions collapse with the quasi self-similar ring profile, with ring width L(t) that vanishes at singularity, and radius proportional to L^α, where α=(4-σ)/(σ(d-1)). The blowup rate of these solutions is 1/(3+α) for 4/d\leσ<4, and slightly faster than 1/4 for σ=4. These solutions are analogous to the ring-type solutions of the nonlinear Schrodinger equation.

On the inhomogeneous biharmonic nonlinear Schrödinger equation: local, global and stability results

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $$ i u_t +Δ^2 u+λ|x|^{-b}|u|^αu = 0, $$ where $λ=\pm 1$ and $α$, $b>0$. We show local and global well-posedness in $H^s(\mathbb{R}^N)$ in the $H^s$-subcritical case, with $s=0,2$. Moreover, we prove a stability result in $H^2(\mathbb{R}^N)$, in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates related to the linear problem.

Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity

We consider positive solutions $u$ of the semilinear biharmonic equation $Δ^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a periodic function of $\ln |x|$ and we classify all such solutions.

Normalized ground states of nonlinear biharmonic Schrödinger equations with Sobolev critical growth and combined nonlinearities

This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} Δ^{2}u-λu=μ|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $μ>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ is the $H^2$-critical Sobolev exponent, and $λ$ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are p…

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 $$ Δ^2 u = u^p~~~~~~~in ~ B_1 \backslash \{0\}$$ with an isolated singularity, where the punctured ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n\geq 5$ 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 $-Δu$. We also give a new method to prove removable singul…

指数非线性增长下涉及连续势函数的加权双调和方程

Weighted biharmonic equations involving continous potentiel under exponential nonlinear growth

本文研究四维单位球中含对数权重与连续正势的临界指数增长双调和方程，利用集中紧性与渐近条件克服紧性缺失，并通过山路定理证明非平凡正弱解的存在性。

Blowing up Solutions for a Biharmonic Equation with Critical Nonlinearity

In this paper we consider the following biharmonic equation with critical exponent $P_ε$ : $Δ^2 u= Ku^{(n+4)/(n-4)-ε}, u>0$ in $Ω$ and $u=Δu=0$ on $\partialΩ$, where $Ω$ is a domain in $R^n$, $n\geq 5$, $ε$ is a small positive parameter and $K$ is smooth positive function. We construct solutions of $P_ε$ which blow up and concentrate at strict local maximum of $K$ either at the boundary or in the interior of $Ω$. We also construct solutions of $P_ε$ concentrating at an interior strict local minimum of $K$. Finally, we prove a nonexistense result for the corresponding supercritical problem whic…

The Gel'fand problem for the biharmonic operator

We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains.

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 equation $$ Δ^{2} u -Δu+(λa+1)u= |u|^{p-2}u $$ 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 $u_λ$. Moreover, we prove that as $λ\rightarrow +\infty$, the solutions $u_λ$ converge to a solution of the equation \begin{align*} \begin{cases} Δ^{2}u -Δu+u = |u|^{p-2}u, &\text…

Nonexistence of Positive Supersolutions of Nonlinear Biharmonic Equations without the Maximum Principle

We study classical positive solutions of the biharmonic inequality $-Δ^2 v \geq f(v)$ in exterior domains in $\mathbb{R}^n$ where $f:(0,\infty)\to (0,\infty)$ is continuous function. We give lower bounds on the growth of $f(s)$ at $s=0$ and/or $s=\infty$ such that this inequality has no $C^4$ positive solution in any exterior domain of $\mathbb R^n$. Similar results were obtained by Armstrong and Sirakov [ Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011-2047] for $-Δv\ge f(v)$ using a method which depe…

Biharmonic equation with singular nonlinearity

We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & Δ^2 u = K(x)u^{-α} \quad \mbox{ in }\,Ω, \\ &u> 0\quad \mbox{ in }\,Ω, \;\;u\vert_{\partialΩ}=0, \,Δu\vert_{\partialΩ} = 0. \end{array}\right. \end{eqnarray*} We prove the main existence result: Assume that $α+β<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \begin{eqnarray}\label{behaviour-bound} c_1 ρ(x)\leq u(x)\leq c_2 ρ(x) \end{eqnarray} where $ρ(x)=d(x,\partialΩ)$. This result is sharp: Assume that $α+β\geq 2$. Then, there is no s…

Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaces

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $d\in \mathbb N$, $s\ge 0$, $0<b<4$, $σ>0$ and $λ\in \mathbb R$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in $H^{s}(\mathbb R^{d})$ if $d\in \mathbb N$, $0\le s <\min \{2+\frac{d}{2},\frac{3}{2}d\}$, $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$ and $0<σ< σ_{c}(s)$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c…

Orlicz-Sobolev空间中双调和算子的非线性特征值问题

Nonlinear eigenvalue problems for a biharmonic operator in Orlicz-Sobolev spaces

本文在Orlicz-Sobolev空间中提出推广的双调和g-拉普拉斯算子Δ_g²，建立其泛函性质，研究依赖归一化条件的特征值及相关非线性问题的谱分布特性。

Asymptotic expansion of radial solutions for supercritical biharmonic equations

Consider the positive, radial solutions of the nonlinear biharmonic equation $Δ^2 u = u^p$. There is a critical power $p_c$ such that solutions are linearly stable if and only if $p\geq p_c$. We obtain their asymptotic expansion at infinity in the case that $p\geq p_c$.

四阶方程中带加权指数非线性项的对称破坏

Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities

本文研究了四阶方程中带加权指数非线性项的对称破坏问题，通过建立加权Adams不等式并证明当权重指数较大时极值点的径向对称性被破坏，扩展了经典结果到四阶PDEs领域。

Energy scattering for a class of inhomogeneous biharmonic nonlinear Schrödinger equations in low dimensions

We consider a class of biharmonic nonlinear Schrödinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -Δ^2 u+μΔu +|x|^{-b} |u|^αu=0, \quad \left. u\right|_{t=0}=u_0 \in H^2(\mathbb{R}^d) \] with $d\geq 1, μ\geq 0$, $0<b<\min\{d,4\}$, $α>0$, and $α<\frac{8-2b}{d-4}$ if $d\geq 5$. We first determine a region in which solutions to the equation exist globally in time. We then show that these global-in-time solutions scatter in $H^2(\mathbb{R}^d)$ in three and higher dimensions. In the case of no harmonic perturbation, i.e., $μ=0$, our result extends the energy sc…

Picone's Identity for $p$-biharmonic operator and Its Applications

In this article we prove the nonlinear analogue of Picone's identity for $p-$biharmonic operator. As an application of our result we show that the Morse index of the zero solution to a $p-$biharmonic boundary value problem is $0$. We also prove a Hardy type inequality and Sturmian comparison principle. We also show the strict monotonicity of the principle eigenvalue and linear relationship between the solutions of a system of singular $p$-biharmonic system.

A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities

We study the existence and non-existence of nontrivial weak solution of $$ {Δ^2u-μ\frac{u}{|x|^{4}} = \frac{|u|^{q_β-2}u}{|x|^β}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0<β<4$, $1<q\leq 2^{**}$ and $μ<μ_1:=\big(\frac{N(N-4)}{4}\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\leqμ<μ_1$, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz…

低维双调和非线性薛定谔方程的局部适定性

Local well-posedness for the biharmonic nonlinear Schrödinger equation in low dimensions

本文证明了低维（1、2、3维）四阶双调和非线性薛定谔方程在加权Sobolev子集及H^s空间中的局部适定性，为光学中四次孤子的数学理论奠定基础。

Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Let $λ^{*}>0$ denote the largest possible value of $λ$ such that $$ \{{array}{lllllll} Δ^{2}u=λ(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. $$ has a solution, where $\B$ is the unit ball in $R^{n}$ centered at the origin, $p>\frac{n+4}{n-4}$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the uni…

Singular solutions of the biharmonic Nonlinear Schrodinger equation

We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile…

An inverse random source problem for the biharmonic wave equation

This paper is concerned with an inverse source problem for the stochastic biharmonic operator wave equation. The driven source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The well-posedness of the direct problem is examined in the distribution sense and the regularity of the solution is discussed for the given rough source. For the inverse problem, the strength of the random source, involved in the principal symbol of its covariance operator, is shown to be uniquely determined by a single realizati…

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

In this work, we investigate the 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 $ Ξ(z)=e^{|z|^p}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution …

Combined effects for non-autonomous singular biharmonic problems

We study the existence of nontrivial weak solutions for a class of generalized $p(x)$-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.

Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity

In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ Δ^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.

具有组合非线性项的双调和非线性薛定谔方程的归一化解

Normalized solutions to the biharmonic nonlinear Schrödinger equation with combined nonlinearities

本文研究具有组合非线性项的双调和非线性薛定谔方程在给定质量约束下归一化基态解的存在性，利用Sobolev次临界逼近方法证明了临界情形下第二山路型临界点的存在，并讨论了参数μ的不同符号情形。

The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearit…