Cristina Pocci
In this paper we use the method of matched asymptotic expansions in order to obtain a geometric motion as the singular limit of a nonlinear fourth order inhomogeneous equation.
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A note on the fourth-order Schrodinger equation with spatially growing inhomogeneous source term
本文研究了一个非线性双调和薛定谔方程，考虑了非均匀项的影响，发展了小数据的局部和全局理论，并在更低正则性下发展了局部理论。
Blow-up results for Inhomogeneous fourth-order nonlinear Schrödinger Equation
本文研究非齐次双调和薛定谔方程解的H²范数爆破现象，通过凸性方法和virial恒等式，分别分析负能量及低于基态能量且动能超基态的正能量下径向与非径向解的爆破情况。
We study the $L^1-L^\infty$ dispersive estimate of the inhomogeneous fourth-order Schrödinger operator $H=Δ^{2}-Δ+V(x)$ with zero energy obstructions in $\mathbf{R}^{3}$. For the related propagator $e^{-itH}$, we prove that for $0<t\leq 1$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/4}$-estimate. For $t>1$, we prove that:\,\, 1) if zero is a regular point of $H$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 2) if zero is a resonance of $H$, there exists a time dependent operator $F_{t}$ such that $e^{-itH}P_{ac}(H)-F_{t}$ satisfies the $|t|^{-3/2}$- disper…
Non-radial Blow-up for a mass-critical fourth-order inhomogeneous nonlinear Schrödinger equation
本文研究了质量临界四阶非齐次非线性薛定谔方程的非径向解在负能量下的有限或无限时间破裂现象。
We consider a free boundary problem modeling electrostatic microelectromechanical systems. The model consists of a fourth-order damped wave equation for the elastic plate displacement which is coupled to an elliptic equation for the electrostatic potential. We first review some recent results on existence and non-existence of steady-states as well as on local and global well-posedness of the dynamical problem, the main focus being on the possible touchdown behavior of the elastic plate. We then investigate the behavior of the solutions in the time singular limit when the ratio between inertial…
This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Ampère problem in the $n$-dimensional radial symmetric case. A detailed study of the fourth order problem is presented. In particular, various {\em a priori} estimates with explicit dependence on the perturbation parameter $\vepsi$ are derived, and a crucial convexity property is also proved for the solution of the fourth order problem. Using these estimates and the convexity property, we prove that the solution of the perturbed problem converges uniformly and compactly to the uniqu…
Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term
本文研究了分数半线性热方程中带有奇异非齐次项的解的存在性，通过建立分数热半群的衰减估计和实插值法，获得了精确的积分估计和存在条件。
We study the nonlinear inhomogeneous wave equation in one space dimension: $v_{tt} - T(v,x)_{xx} = 0$. By constructing some "decoupled" Riccati type equations for smooth solutions, we provide a singularity formation result without restrictions on the total variation of unknown, which generalize earlier singularity results of Lax and the first author. These results are applied to several one-dimensional hyperbolic models, such as compressible Euler flows with a general pressure law, elasticity in an inhomogeneous medium, transverse MHD flow, and compressible flow in a variable area duct.
Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the ra…
We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge-Ampère type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu type equations. Both the Legendre transform and partial Legendre transform are used in our analysis. In two dimensions, we establish global solutions to the second boundary value problem for highly singular Abreu equations where the right hand sides are of $q$-Laplacian type for all $q>1$. We show that minimizers of variational problems with a convexity constr…
A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a-priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is als…
This manuscript proves the energy scattering of global solutions to a repulsive fourth-order generalized Hartree equation with non-radial data in the inter-critical regime. This work uses a new approach due to Dodson-Murphy [4] and extends the previous work [14] by removing the spherically symmetric assumption on the data.
In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation $$\{\begin{array}{lllllll} βΔ^{2}u-τΔu=\fracλ{(1-u)^{p}} & in\ \ B, 0<u\leq 1 & in\ \ B, u=Δu=0 & on\ \ \partial B, \end{array} . $$ where $B$ is the unit ball in $R^{n}$, $λ>0$ is a parameter, $τ>0, β>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.}
In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=Δ^2-V=Δ^2-c|x|^{-4}$ where $c$ is any constant such that $c<\left(\frac{N(N-4)}{4}\right)^2$. The semigroup generated by $-A$ in $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $ΔA^{-1/2}$ on $L^p(\mathbb{R}^N)$ for all $p\in(p_0^{'},2]$.
This paper is devoted to the study of the inhomogeneous wave equation with singular (less than continuous) time dependent coefficients. Particular attention is given to the role of the lower order terms and suitable Levi conditions are formulated in order to obtain a very weak solution as introduced in \cite{GR:14}. Very weak solutions for this kind of equations are also investigated from a numerical point of view in two toy models: the wave equation with a Heaviside function and a delta distribution, respectively, as coefficient in its principal part.
This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$i\dot u+Δ^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson-Murphy \cite{dm}.
We consider a singularly perturbed fourth-order problem with third-order terms on the unit square. With a formal power series approach, we decompose the solution into solutions of reduced (third-order) problems and various layer parts. The existence of unique solutions for the problem itself and for the reduced third-order problems is also addressed.
Using the method of Nehari manifold, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singulatities and with critical Sobolev growth.
We show the existence and the regularity properties of the weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient, by analyzing a fourth-order nonlinear elliptic equation for the stream function. The density function and the viscosity coefficient may have large variations. In addition, we formulate the solutions for the parallel, concentric and radial flows respectively, and as examples we calculate the solutions with piecewise-constant viscosity coefficients explicitly.
The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity
针对强奇性三维能量临界 INLS 的未解 Cauchy 问题，论文改进加权空间上的非齐次 Strichartz 估计，建立了 $3/2\leq α<11/6$ 范围内的局部及小数据全局适定性。
Sharp regularity estimates for a singular inhomogeneous (m, p)-Laplacian equation
本文针对一类双重非线性发展偏微分方程，结合几何切向方法与内蕴尺度技术建立解在Hölder空间的精确正则性，统一处理三类扩散情形推广经典结果，并给出相关非线性模型的应用。
Local behavior of solutions to inhomogeneous singular parabolic $p$-Laplace equations
本文通过时间重标度与$L^\infty$比较估计，将非齐次奇异抛物$p$-Laplace方程解的正则性证明归结为空间水平集测度的椭圆型衰减估计，从而克服了内蕴尺度方法中的主要困难。
The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $λ$. It is shown that there is a threshold value $λ_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $λ>λ_*$, while at least two such solutions exist for $λ\in (0,λ_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evol…
We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it anisotropic} homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission prob…
Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we pre- dict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two …
We prove some qualitative properties for singular solutions to a class of strongly coupled system involving a Gross--Pitaevskii-type nonlinearity. Our main theorems are vectorial fourth order counterparts of the classical results of [J. Serrin, Acta Math. (1964)], [P.-L. Lions, J. Differential Equations (1980)], [P. Aviles, Comm. Math. Phys. (1987)], and [B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1981)]. On the technical level, we use the moving sphere method to classify the limit blow-up solutions to our system. Besides, applying asymptotic analysis techniques, we show that these soluti…
We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor blades (or persistent singularities).
Stationary states of a chemotaxis consumption system with singular sensitivity and inhomogeneous boundary conditions
本文证明了具有奇异灵敏度的稳态趋化-消耗模型在二维有界光滑区域及三维以上球体中，对给定总质量存在唯一解。
In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Ampère type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial Legendre transform. As an application, we obtain a new proof of the Bernstein theorem without using Caffarelli-Gutiérrez's estimate, including the Chern conjecture on affine maximal surfaces. For the inhomogeneous equation, we also obtain a new proof in dimension two by an integral method relying on the Monge-Ampère Sobolev inequality. This proof works even when…
Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential
本文针对具有强奇异势的四阶 Baouendi-Grushin 型次椭圆方程，通过引入 Almgren 型频率函数并证明其单调性，结合带边界项的 Hardy-Rellich 型不等式，建立了强唯一延拓性。
A one-way wave equation is an evolution equation in one of the space directions that describes (approximately) a wave field. The exact wave field is approximated in a high frequency, microlocal sense. Here we derive the pseudodifferential one-way wave equation for an inhomogeneous acoustic medium using a known factorization argument. We give explicitly the two highest order terms, that are necessary for approximating the solution. A wave front (singularity) whose propagation velocity has non-zero component in the special direction is correctly described. The equation can't describe singulariti…
We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_νu=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0<u_\la<1$ exists for $\la\in (0,\la^*)$, while there is none of any kind when $\la>\la^*$. Our main result asserts that the extremal solution $u_{…
Self-similar large time behaviour of weak solutions of the fourth-order parabolic thin film equations with absorption is studued.
We study the asymptotic behavior for singular solutions to a critical fourth order system generalizing the constant $Q$-curvature equation. Our main result extends to the case of strongly coupled systems, the celebrated asymptotic classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] and [N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Invent. Math., (1999)]. On the technical level, we use an involved spectral analysis to study the Jacobi fields' growth properties in the kernel of the linearization of our system around a blow-up limit solution. Beside…
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
Continuous branches of similarity solutions of the fourth-order stable thin film equation are studied.
We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and stro…
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…
Convergence in the incompressible limit of the corner singularities
本文研究消去压力后的 Stokes 惩罚系统在不可压缩极限下的角点奇异性，建立 Lamé 与 Stokes 问题奇异系数公式并证明其收敛性。
Boundary controllability for a fourth order degenerate parabolic equation with a singular potential
本文研究带奇异势的一维四阶退化抛物型方程在左端点边界控制下的零能控性，利用 Bessel 函数谱分解、半群理论和矩方法给出能控性代价的上下界估计。
So far most studies on mathematical models for microelectromechanical systems (MEMS) are focused on the so-called small aspect ratio model which is a wave or beam equation with a singular source term. It is formally derived by setting the aspect ratio equal to zero in a more complex model involving a moving boundary. A rigorous justification of this derivation is provided here when bending is taken into account.
A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions
本文提出了一种四阶正则化的浸入平面曲线曲率流，通过弹性体类型功能证明其在趋近于0时收敛到具有Dirichlet边界条件的曲率流。
A new method to compute Schauder Estimates for multidimensional fourth order heat-type equations is proposed. In particular, we show how knowing Schauder or Sobolev estimates for the one-dimensional fourth order heat equation allows to derive their multidimensional analogs for equations with time inhomogeneous coefficients with the same constants as in the case of the one-dimensional heat equation. Our method relies on a merger between (Krylov and Priola, 2017), where they actually showed the same result for the classical second order heat equation and (Funaki, 1979), where a probabilistic con…
We consider a fourth order evolution equation involving a singular nonlinear term $\fracλ{(1-u)^{2}}$ in a bounded domain $Ω\subset\R^{n}$. This equation arises in the modeling of microelectromechanical systems. We first investigate the well-posedness of a fourth order parabolic equation which has been studied in \cite{Lau}, where the authors, by the semigroup argument, obtained the well-posedness of this equation for $n\leq2$. Instead of semigroup method, we use the Faedo-Galerkin technique to construct a unique solution of the fourth order parabolic equation for $n\leq7$, which improves and …
Threshold for the existence of scattering states for inhomogeneous nonlinear Schrödinger equations without gauge invariance
本文研究非规范不变非齐次非线性薛定谔方程，在强制性条件下证明低于Strauss型阈值时无散射态，并利用Lorentz空间中的非容许Strichartz估计证明阈值以上的小数据散射结果。
Fourth order Hardy-Sobolev equations: Singularity and doubly critical exponent
本文研究了$\mathbb{R}_+^N$上含双重临界指数的四阶Hardy-Sobolev方程，利用Mountain-Pass定理及新证的Hardy-Sobolev嵌入极值存在性，证明了在特定维度和参数条件下非平凡弱解的存在性。
We study the regularity of the extremal solution of the semilinear biharmonic equation $βΔ^2 u-τΔu=\fracλ{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=Δu=0$ on $\partial B$, where $λ>0$ is a parameter, while $τ>0$, $β>0$ are fixed constants. It is known that there exists a $λ^{*}$ such that for $λ>λ^{*}$ there is no solution while for $λ<λ^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and $β, τ>0$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$, $β>0$, and $τ…
Inhomogeneous incompressible Euler with codimension $1$ singular structures
本文研究非齐次不可压缩Euler方程，建立能量通量的Duchon--Robert型逼近定理，证明余维1奇异界面上耗散测度为零的结论，拓展了不可压缩流体能量耗散的理论结果。
Non-local parabolic equations with singular (Morrey) time-inhomogeneous drift
该研究针对带局部无界奇异漂移项的非局部抛物方程，借助Adams和Krylov的思路证明抛物算子范数不等式，得到解的Sobolev正则性估计，同时给出对应Feller稳定过程的Krylov界与McKean-Vlasov方程解的先验正则性估计。
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.
Inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities
研究不均匀非线性薛定谔方程的 stationary 状态所对应的非线性椭圆方程，通过加权 Sobolev 空间嵌入和临界点理论，获得非线性特征值的存在性和多重性结果，并利用 Pohozaev 型恒等式证明非存在性结果。
In this paper we study the following singular perturbation problem for the $p_\varepsilon(x)$-Laplacian: \[ Δ_{p_\varepsilon(x)}u^\varepsilon:=\mbox{div}(|\nabla u^\varepsilon(x)|^{p_\varepsilon(x)-2}\nabla u^\varepsilon)=β_{\varepsilon}(u^\varepsilon)+f_\varepsilon, \quad u^\varepsilon\geq 0, \] where $\varepsilon>0$, $β_{\varepsilon}(s)={1 \over \varepsilon} β({s \over \varepsilon})$, with $β$ a Lipschitz function satisfying $β>0$ in $(0,1)$, $β\equiv 0$ outside $(0,1)$ and $\int β(s)\, ds=M$. The functions $u^\varepsilon$, $f_\varepsilon$ and $p_\varepsilon$ are uniformly bounded. We prove …
In this paper, we study some fourth order singular critical equations of Lichnerowicz type involving the Paneitz-Branson operator, and we prove existence and non existence results under given assumptions.
The necessary and sufficient conditions for a regular positive entire solution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -Δ^2 u=u^{-p} \;\; \mbox{in $\R^N \; (N \geq 3)$}, \;\; p>1 \end{equation} to be a radially symmetric solution are obtained via the moving plane method (MPM) of a system of equations. It is well-known that for any $a>0$, \eqref{0.1} admits a unique minimal positive entire radial solution ${\underline u}_a (r)$ and a family of non-minimal positive entire radial solutions $u_a (r)$ such that $u_a (0)={\underline u}_a (0)=a$ and $u_a (r) \geq {\underline u}_a…
We obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are induced by the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequalitry and an interpolation inequality, which are related to the singula…
In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic syste…
We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving s…
We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by establishing the refined analysis of the spatially decaying nonlinear term.
We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we show that for any solution with a finite time singularity, the $L^{\infty}$ norm of the curvature must go to infinity. Furthermore, we characterize the behavior at infinity of solutions with bounded curvature.
The well known heat equation with finite speed of propagation proposed by Cattaneo is obtained by a more general fourth order PDE when a certain (small) parameters is put equal to zero. It seems that this fourth order equation has been essentially overlooked in the literature, in particular when it is subject to non homogeneous boundary conditions. In this paper we examine its well posedness and the asymptotic behavior when certain coefficients tend to singular values.
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