We construct a solution to a complex nonlinear heat equation which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. We note that the real and imaginary parts of the constructed solution blow up simultaneously.
We consider convex potentials $W:\R\to [0,\infty)$ vanishing at $0$ and growing sufficiently fast at $\pm\infty$. Given any open set $Ω\subset\R^n$ with Lipschitz and compact boundary, we prove the existence and uniqueness of a solution of $Δu= W'(u)$ in $Ω$, such that $u=+\infty$ or $u=-\infty$ on $\partial Ω$. Moreover, if $\partial Ω$ is the union of two disjoint compact subsets $A^+$ and $A^-$, there also exists a unique solution satisfying $u=+\infty$ on $A^+$ and $u=-\infty$ on $A^-$.
In the present paper initial problems for the semilinear integro-differential diffusion equation and system are considered. The analogue of Duhamel principle for the linear integro-differential diffusion equation is proved. The results on existence of local mild solutions and Fujita-type critical exponents to the semilinear integro-differential diffusion equation and system are presented.
Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be type II in critical case. When p>p_S=(N+2)/(N-2) for N>=3, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/(p-1), which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collaps…
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
This paper is concerned with finite blow-up solutions of a one dimensional complex-valued semilinear heat equation. We provide locations and the number of blow-up points from the viewpoint of zeros of the solution.
We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighborhood of its minimum. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential.
We consider the semilinear heat equation in large dimension $d\geq 11$ $$ \partial_t u =Δu+|u| ^{p-1}u, \ \ p=2q+1, \ \ q\in \mathbb N $$ on a smooth bounded domain $Ω\subset \mathbb R^d$ with Dirichlet boundary condition. In the supercritical range $p\geq p(d)>1+\frac{4}{d-2}$ we prove the existence of a countable family $(u_\ell)_{\ell \in \mathbb N}$ of solutions blowing-up at time $T>0$ with type II blow up: $$ \parallel u_{\ell}(t) \parallel_{L^{\infty}} \sim C (T-t)^{-c_\ell} $$ with blow-up speed $c_\ell>\frac{1}{p-1}$. They concentrate the ground state $Q$ being the only radially and d…
In this paper we study the so-called large solutions of elliptic semilinear equations with non null sources term, thus solutions blowing up on the boundary of the domain for which reason they are greater than any other solution whenever Weak Maximum Principle holds. The main topic about large solutions is uniqueness results and their behavior near the boundary. It is much less than being simple. The structure of the semilinear equations considered includes the well known Keller-Osserman integral and an assumption on the ellipticity of the leading part of the differential operator. In our study…
We consider a vector-valued blow-up solution with values in $\mathbb{R}^m$ for the semilinear wave equation with power nonlinearity in one space dimension (this is a system of PDEs). We first characterize all the solutions of the associated stationary problem as an m-parameter family. Then, we show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic cases. Our analysis is not just a simple adaptation of the already handled real or complex case. In particular, there is a new structure of the set a stationary solutio…
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real ca…
We verify the existence of radial positive solutions for the semi-linear equation $$ -\,Δu=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a radial smooth potential. If $q$ is super-critical, namely $q>p^*$, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with different rates of concentration, provided $V(0)<0$. On the other hand, if $N/(N-2)<q<p^*$, we prove that this Problem has a radial solution behaving like a super-position of {\it …
Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify th…
In this paper, we consider the standard semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + |u|^{p-1}u, \quad p >1. \end{eqnarray*} The determination of the (believed to be) generic blowup profile is well-established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of…
We prove local in time well-posedness in Sobolev spaces of the Cauchy problem for semi-linear p-evolution equations of the first order with real principal part, but complex valued coefficients for the lower order terms, assuming decay conditions on the imaginary parts as |x| goes to infinity.
We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\in H^r_{\per}(\R)$ for $r\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\ast$ such that for $p\in (1,p^\ast)$ both in the "+" and the "-" case we can us…
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points $I_0$ is open and that the blow-up curve is of class $C^{1,μ_0}$ and the phase $θ$ is $C^{μ_0}$ on this set. In order to prove this result, we introduce a Liouville Theorem for that equation.
Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation $$ -Δu-u=Q|u|^{p-1}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim_{|x|\to0}u(x)=+\infty, $$ where $N\geq 2$, $p>1$ and the potential $Q: \mathbb{R}^N\to (0,+\infty)$ is a Hölder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin's subcritical case and then isolated singular solutions is derived with the form $u_k=kΦ+v_k$ via the Schauder fixed point theorem for the integral equation $$v_k=Φ\ast\big…
We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that mild solutions exist and are unique in suitably chosen functional classes. More precisely, function-valued solutions are obtained, as well as a regularity result.
We focus on the study of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo-Nirenberg inequality and we study global existence, $L^2$-concentration phenomena and blowup profile for the ev…
In this paper, we investigate the semilinear equation with a time-space fractional structural damping and a nonlocal in time nonlinearity \begin{equation*} {\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^σu+(-Δ)^δ\mathbf{D}_{0|t}^{α_2} u = I_{0|t}^{1-γ}|u|^{p}, \qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \end{equation*} where $p>1$, $α_i, γ in (0,1)$, $δ, σ\in (0,1)$, ${\mathbf{D}}_{0|t}^{α_i}$ is the Caputo fractional derivative and $I_{0|t}^{1-γ}$ is the Riemann-Liouville fractional integral of order $1-γ$. We prove the non-existence of global solutions if \begin{equation*} 1<p\leqslant \frac{2(2+α_1…
Generically sharp decay and blowing up at infinity for a weak null wave system
Southern University of Science and Technology（南方科技大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Xi’an Jiaotong University（西安交通大学）
We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-Δu+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of wavelength $\sim h$ and amplitude $\sim 1$. They propagate in a regime where the non-linearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits $\text{supp}_x f$. We show that one can recover $f(x,u)$ when it is an odd function of $u$, and we can recover $α(x)$ when $f(x,u)=α(x)u^{2m}…
In this paper, we consider the nonlinear Schrödinger equation with a real valued potential V=V(x). We study global behavior of solutions to the equation with a data below the ground state under some conditions for the potential V and prove a scattering result and a blowing-up result in mass-supercritical and energy-subcritical. Our proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du-Wu-Zhang in [6]. We can exclude the possibility of the growing-up result by the argument in [23], [15], and [10] if "the data and the potential are radi…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that these mild solutions exist uniquely in suitably chosen functional classes.
We consider the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0 \mbox{ for } (x,t)\in \mathbb{R}^3\times\mathbb{R}$. For any $p>1$ we prove the existence of time-periodic spatially localized real-valued solutions (breathers) both for the $+$ and the $-$ case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to $0$ as $|x|\to \infty$. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the …
This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the Hölder continuity with a non-optimal exponent.
We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on $L_p$ spaces, driven by multiplicative Wiener noise, with a drift term given by an evaluation operator that is assumed to be quasi-monotone and polynomially growing, but not necessarily continuous. In particular, we consider a notion of mild solution ensuring that the evaluation operator applied to the solution is still function-valued, but satisfies only minimal integrability conditions. The proofs rely on stochastic calculus in Banach spaces, monotonicity and convexity techniques, and we…
We study the semilinear elliptic equation --$Δ$u + g(u)$σ$ = $μ$ with Dirichlet boundary condition in a smooth bounded domain where $σ$ is a nonnegative Radon measure, $μ$ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that $σ$ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| ^{q--1} r, with q > 1 and $μ$ satisfies an absolute continuity condition expressed in terms of s…
We prove existence results of complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory.
We study the Cauchy problem for Schrödinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.
We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and of the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path-dependent. The solution i…
We discuss a notion of weak solution for a semilinear wave equation that models the interaction of an elastic body with a rigid substrate through an adhesive layer, relying on results in [2]. Our analysis embraces the vector-valued case in arbitrary dimension as well as the case of non-local operators (e.g. fractional Laplacian).
In this paper we study the semilinear elliptic problem $$ -Δu -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2, $$ where $k>0$, $p\geq 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for decaying and for periodic coefficient $Q$. In addition, a nonlinear far-field relation is derived for these solutions.
We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation $u_{tt}-Δu+ α(x) |u|^2u=0$, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength $h$ and amplitude $h^{-1/2}$, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits the support of $α$. We show that one can extract the Radon transform of $α$ from the phase shift of such waves, and then one can recover $α$. We also show that one can probe the medium with real-valued harmonic waves and obtain uniquen…
In this paper, we consider the following complex-valued semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C}, \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $ p \in \mathbb{N}, p \geq 2$. We aim at constructing for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a$, with the following estimates for the final profile \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}}, u_2(x,T) &\sim & \frac{2 p}{(p-1)^2} \left[ \frac{ (p-1)^2|x-a|^2}{…
Sharp spectral-scale stability for parabolic equations with measure-valued delay
College of Computing, Georgia Institute of Technology（佐治亚理工学院计算学院）；Department of Mathematics and Systems Engineering, Florida Institute of Technology（佛罗里达理工学院数学与系统工程系）
We study the diffusion-reaction-advection model for mobile chemical species together with the dissolution and precipitation of immobile species in a porous medium at the micro-scale. This leads to a system of semilinear parabolic partial differential equations in the pore space coupled with a nonlinear ordinary differential equation at the grain boundary of the solid matrices. The fluid flow within the pore space is given by unsteady Stokes equation. The novelty of this work is to do the iterative limit analysis of the system by tackling the nonlinear terms, monotone multi-valued dissolution r…
We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution $u=0$ is the only solution for which the assumptions $u(t=0)\vert_{D}=0, u(t=T)\vert_{D}=0$ hold, where $D\subset \mathbb{R}^d$ are certain subsets of codimension one. In particular, $D$ is discrete for dimension $d=1$. Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko--Viazovska formula, and the uniqueness res…
Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations
Cadi Ayyad University（卡迪·阿亚德大学）；Faculty of Sciences Semlalia, Laboratory of Mathematics, Modeling and Automatic Systems（塞姆拉利亚理学院，数学、建模与自动系统实验室）；The Vanguard Center, University Mohammed VI Polytechnic（穆罕默德六世理工大学先锋中心）；Faculty of Sciences-Agadir, Lab-SIV, Ibn Zohr University（阿加迪尔理学院，SIV实验室，伊本·佐赫尔大学）；Department of Mathematics, Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数理科学研究科数学系）；Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University（宗古尔达克布伦特·埃切维特大学理学院数学系）
