We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave equation by modifying Hörmander's method. Next, we construct an approximate solution, by solving our new reduced system given some scattering data at infinite time. Finally, we prove that the quasilinear wave equation has a global solution which agrees with the approximate solution at infinite time.
The authors show that bilinear estimates for null forms hold for Dirichlet-wave equations outside of convex obstacle. This generalizes results for the Euclidean case of Klainerman and Machedon, and of Sogge for the variable coefficient boundaryless case. The estimates are used to prove a local existence theorem for semilinear wave equations satisfying the null condition.
We are interested in the three-dimensional quasilinear wave equations with null condition. Global existence and pointwise decay for this model have been proved in the celebrated works of Klainerman \cite{Klainerman86} and Christodoulou \cite{Christodoulou86} for small smooth initial data. In this work, we illustrate the precise pointwise asymptotic behavior of the solutions for initial data posed on a hyperboloid and show that the decay rate $v^{-1}u^{-1}$ is optimal for a generic set of initial data.
We explore the global existence of solutions to systems of quasilinear wave equations satisfying the null condition when the initial data are sufficiently small. We adapt an approach of Keel, Smith, and Sogge, which relies on integrated local energy estimates and a weighted Sobolev estimate that yields decay in $|x|$, by using the $r^p$-weighted local energy estimates of Dafermos and Rodnianski. One advantage of this approach is that all time-dependent vector fields can be avoided and the proof can be readily adapted to address wave equations exterior to star-shaped obstacles.
We prove global existence of solutions to multiple speed, Dirichlet-wave equations with quadratic nonlinearities satisfying the null condition in the exterior of compact obstacles. This extends the result of our previous paper by allowing general higher order terms. In the currect setting, these terms are much more difficult to handle than for the free wave equation, and we do so using an analog of a pointwise estimate due to Kubota and Yokoyama.
In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu's seminal work [G. Luli, S. Yang, P. Yu, On one-dimension semi-linear wave equations with null conditions, Adv. Math.329 (2018) 174-188] from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense. In order to achieve these goals, we will employ Luli, Yang and Yu's weighted energy estimates with positive weights, intr…
We consider a class of quasilinear wave equations in $3+1$ space-time dimensions that satisfy the "weak null condition" as defined by Lindblad and Rodnianski \cite{LR1}, and study the large time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is $-\partial_t^2 u + (1+u) Δu = 0$. Global solutions for such equations have been constructed by Lindblad \cite{Lindblad1,Lindblad2} and Alinhac \cite{Alinhac1}. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solution…
We consider the two-dimensional quasilinear wave equations with quadratic nonlinearities. We introduce a new class of null forms and prove uniform boundedness of the highest order norm of the solution for all time. This class of null forms include several prototypical strong null conditions as special cases. To handle the critical decay near the light cone we inflate the nonlinearity through a new normal form type transformation which is based on a deep cancelation between the tangential and normal derivatives with respect to the light cone. Our proof does not employ the Lorentz boost and can …
In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time $L^2$ estimates.
We establish global existence in 3+1 dimensions of small-amplitude solutions of quasilinear Dirichlet-wave equations satisfying the null condition outside of star-shapped obstacles.
We consider the two-dimensional quasilinear wave equations with standard null-form type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.
We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of "natural constraint"
We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p-${L}aplacian operator with a coefficient that is {locally bounded inside the domain and satisfying certain additional integrability assumptions}. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value pro\-blem …
We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation. Thus, we can handle all non-trapping obstacles. We are also able to handle non-diagonal systems satisfying the appropriate null condition.
For the 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(\nabla u)\partial_{ij}u=0$ with coefficients independent of the solution $u$, a blowup result for small data solutions has been established in [1,2] provided that the null condition does not hold and a generic nondegeneracy condition on the initial data is fulfilled. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(u, \nabla u)\partial_{ij}u=0$ with coefficients that depend simultaneously on $u$ and $\nabla u$. When the null condition does not hold an…
Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
School of Mathematical Sciences and IMS, Nanjing Normal University（南京师范大学数学科学学院和集成数学科学研究所）；Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）
A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the exterior domain problems.
The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions.
In this paper we study the following problem. For any $\ep>0$, take $u^{\ep}$ a solution of, $$ Łu^{\ep}:= {div}\Big(\di\frac {g(|\nabla \uep|)}{|\nabla \uep|}\nabla \uep\Big)=β_{\ep}(u^{\ep}),\quad u^{\ep}\geq 0. $$ A solution to $(P_{\ep})$ is a function $u^{\ep}\in W^{1,G}(Ω)\cap L^{\infty}(Ω)$ such that $$ \int_Ω g(|\nabla u^{\ep}|) \frac{\nabla u^{\ep}}{|\nabla u^{\ep}|} \nabla ϕdx =-\int_Ω ϕβ_{\ep}(u^{\ep}) dx $$ for every $ϕ\in C_0^{\infty}(Ω)$. Here $β_{\ep}(s)= \frac{1}{\ep} β(\frac{s}{\ep}), $ with $β\in {Lip}(\R)$, $β>0$ in $(0,1)$ and $β=0$ otherwise. We are interested in the limit…
In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.
A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for $(1+1)$-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.
We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with \textbf{time dependent} inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely $C^1$ close to the Minkowski metric inside some large cylinder $\{\left.(t, x)\right||x|\leq R\}$ and approaches the Minkowski metric weakly as $|x|\rightarrow \infty$. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equati…
We prove global stability for a system of nonlinear wave equations satisfying a generalized null condition. The generalized null condition allows for null forms whose coefficients have bounded $C^k$ norms. We prove both pointwise decay and improved decay of good derivatives using bilinear energy estimates and duality arguments. Combining this strategy with the $r^p$ estimates of Dafermos--Rodnianski then allows us to prove global stability. The proof requires analyzing the geometry of intersecting null hypersurfaces adapted to solutions of wave equations.
We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying Ambrosetti-Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. [5], we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. [1].
We show global existence backwards from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behaviour that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assu…
In the significant work of [2], Alinhac proved the global existence of small solutions for 2D quasilinear wave equations under the null conditions. The proof heavily relies on the fact that the initial data have compact support [22]. Whether this constraint can be removed or not is still unclear. In this paper, for fully nonlinear wave equations under the null conditions, we prove the global well-posedness for small initial data without compact support. Moreover, we apply our result to a class of quasilinear wave equations.
We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on $(R^3, g)$, where the metric $g$ is a small perturbation of the flat metric and approaches the Euclidean metric like $(1+|x|)^{-a}$ with $a>1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in the appendix.
In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(ϕ(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }Ω, u=0 & \ \ \mbox{ on } \partialΩ, \end{cases}$$ where $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$), $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(Ω)$, and $f(x,u)$ is a Carathéodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for $(P)$ in the framework of Musielak-Sobolev spaces. The main difficulty…
We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.
In this paper, we obtain $C^{1,α}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switch…
We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave equation. We instead rely upon a class of Keel-Smith-Sogge estimates for the perturbed wave equation. Using this, a notable simplification is made as compared to previous works concerning wave equations in exterior domains: one no longer needs to distinguish the scaling vector field from the other admissible vector fields.
We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for …
Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions, provided that the initial data are compactly supported and sufficiently small in Sobolev norm. In this work, Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions. A natural question then arises whether the time-growth is a true phenomena, despite the possible conservation of basic energy. Analogous problems are also of central importance for Schrödinger e…
We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scal…
We study some properties of the solutions of (E) $\;-\Gd_p u+|\nabla u|^q=0$ in a domain $\Gw \sbs \BBR^N$, mostly when $p\geq q>p-1$. We give a universal priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the positive solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result in expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete non compact manifolds …
In this paper we prove that the equation $ -( r^αϕ(|u'(r)|)u'(r))' = λr^γf(u(r)), ~0<r<R$, where $α, γ, {\bf{R}}$ are given real numbers, $ϕ: (0, \infty) \to (0, \infty)$ is a suitable twice differentiable function, $λ> 0$ is a real parameter and $f:{\bf{R}}\to{\bf{R}}$ is continuous, admits an infinite sequence of sign-changing solutions satisfying $u'(0) =u(R) =0$. The function $f$ is required to satisfy $tf(t)>0$ for $ t\neq 0$. Our technique explores fixed point arguments applied to suitable integral equations and shooting arguments. Our main result extends earlier ones in the case $ϕ$ is …
We study properties of positive functions satisfying (E) --$Δ$u+m|$\nabla$u| q -- u p = 0 is a domain $Ω$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure $μ$ as boundary data, and these conditions are expressed in terms of Bessel capacities on the boundary. We also study removable boundary singularities and solutions with an isolated singularity on $\partial$$Ω$. The different results depends on two critical exponents for p = p c := N +1 N --1 and for q = q c := N +1 N and on the position of q with respect to…
In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x, \nabla u))&=& μ\quad \text{in} ~Ω, u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*} where $μ$ is a finite signed Radon measure in $Ω$, $Ω\subset \mathbb{R}^n$ is a bounded domain such that its complement $\mathbb{R}^n\backslashΩ$ is uniformly $p$-thick and $\mathcal{A}$ is a Carathéodory vector valued function satisfying growth and monotonic…
We consider the Cauchy problem for quadratic nonlinear Klein-Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.-M. Delort, D. Fang, R. Xue (2004), we show the global existence of asymptotically free solutions if the initial data are sufficiently small in some weighted Sobolev space. Our proof is based on an algebraic characterization of nonlinearities satisfying the null condition.
We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth $p(x)>1$, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for $p(x)$-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two diff…
