For compact, isometrically embedded Riemannian manifolds $ N \hookrightarrow \mathbb{R}^L$, we introduce a fourth-order version of the wave map equation. By energy estimates, we prove an $\textit{a priori}$ estimate for smooth local solutions in the energy subcritical dimension $ n = 1,2$. The estimate excludes blow-up of a Sobolev norm in finite existence times. In particular, combining this with recent work of local well-posedness of the Cauchy problem, it follows that for smooth initial data with compact support, there exists a (smooth) unique global solution in dimension $n = 1,2$. We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms.
In this article we prove the global existence of a unique strong solution to the initial boundary-value problem for a fourth-order exponential PDE. The equation we study was originally proposed to study the evolution of crystal surfaces, and was derived by applying a nonstandard scaling regime to a microscopic Markov jump process with Metropolis rates. Our investigation here finds that compared to the PDE's which use Arhenious rates, (and also have a fourth order exponential nonlinearity) the hyperbolic sine nonlinearity in our equation can offer much better control over the exponent term even…
New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.
In this article, for a fourth-order parabolic equation which is closely related for example to the Cahn-Hilliard equation, we study an inverse source problem by interior data and the continuation of solution from lateral Cauchy data. Our method relies on a Carleman estimate and proves conditional stability for both problems.
We consider the fourth order Schrödinger operator $H=Δ^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable $|t|^{-\frac54}$ decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion $e^{-itH}P_{ac}(H)=t^{-\frac34} A_0+t^{-\frac54}A_1$ where $A_0:L^1\to L^\infty$ is an operator of r…
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 dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…
By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or…
In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation $\partial_tu-Δe^{-Δu}+e^{-Δu}-1=0$. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects \cite{GLLM}. Our investigations reveal that we must control the size of both $\left\| e^{-Δu(x,0)}\right\|_{W^{2,2}(Ω)}$ and $\left\| e^{Δu(x,0)}\right\|_{\infty,Ω}$ suitably to achieve our results. Related results in \cite{GM,LS} were established via the Weiner …
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…
Motivated by experiments and formal asymptotic expansions in the physics literature, Maor and Shachar (J. Elasticity 134 (2019), 149-173) studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics. For thin objects they analysed the scaling of the minimal elastic energy as a function of the thickness. In particular they showed that for maps from a ball of radius h in an oriented Riemannian manifold to Euclidean space, the infimum of a model elastic energy per unit volume scales like the fourth power of h and after rescaling one gets convergence to a qua…
In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive $Q$-curvature conformal prescription problem as the Schoen-Yau positive energy theorem does for the Yamabe problem…
