Search

This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution behaves exactly as in the real-valued case. Namely, up to a rotation in the complex plane, the solution decomposes into a sum of a finite number of decoupled solitons with alternate signs. The main novelty of our proof is a resolution of a complex-valued first order Toda system governing the evolution of the positions and the phases of the solitons., Comment: 48 pages, 2 figures

We consider the semilinear wave equation with a power nonlinearity in the radial case. Given $r_0>0$, we construct a blow-up solution such that the solution near $(r_0,T(r_0))$ converges exponentially to a soliton. Moreover, we show that $r_0$ is a non-characteristic point. For that, we translate the question in self-similar variables and use a modulation technique. We will also use energy estimates from the one dimensional case treated by Merle and Zaag. Of course because of the radial setting, we have an additional gradient term which is delicate to handle. That's precisely the purpose of our paper., Comment: 24 pages

In this paper, we construct a singular standing ring solution of the nonlinear heat in the radial case. We give rigorous proof for the existence of a ring blow-up solution in finite time. This result was predicted formally by Baruch, Fibich and Gavish \cite{BFGpd10}. We also prove the stability of these dynamics among radially symmetric solutions., Comment: 25 pages

We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a $|\log(T-t)|^q$ improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.

We develop a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. In order to numerically reproduce the blow-up phenomena, we propose a rule of scaling transformation, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A careful study of the convergence of the proposed scheme is carried out and several numerical examples are performed in illustration., Comment: 20 pages, 5 figures, 2 tables

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant., Comment: 28 pages

In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i \beta) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u, \quad \text{where } \beta, \delta, \alpha \in \mathbb{R}. $$ The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for \(\beta\) and \(\delta\). Specifically, for a fixed \(\beta \in \mathbb{R}\), the existence of finite-time blow-up solutions for arbitrarily large values of \( |\delta| \) is still unknown. According to a conjecture made by Popp et al. \cite{POPphd98}, when \(\beta = 0\) and \(\delta\) is large, blow-up does not occur for \textit{generic initial data}. In this paper, we show that their conjecture is not valid for all types of initial data, by presenting the existence of blow-up solutions for \(\beta = 0\) and any \(\delta \in \mathbb{R}\) with different types of blowup.

We propose to explore the potential of physics-informed neural networks (PINNs) in solving a class of partial differential equations (PDEs) used to model the propagation of chronic inflammatory bowel diseases, such as Crohn's disease and ulcerative colitis. An unsupervised approach was privileged during the deep neural network training. Given the complexity of the underlying biological system, characterized by intricate feedback loops and limited availability of high-quality data, the aim of this study is to explore the potential of PINNs in solving PDEs. In addition to providing this exploratory assessment, we also aim to emphasize the principles of reproducibility and transparency in our approach, with a specific focus on ensuring the robustness and generalizability through the use of artificial intelligence. We will quantify the relevance of the PINN method with several linear and non-linear PDEs in relation to biology. However, it is important to note that the final solution is dependent on the initial conditions, chosen boundary conditions, and neural network architectures.

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation $$\frac{\partial h}{\partial t}=\frac{\partial^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^\beta},\quad\beta>0$$ subject to initial boundary value conditions $$h(\cdot,0)=h_0>0,\quad h(\pm1,t)=1.$$ We derive the intermediate extinction profile with refined asymptotics, and with extinction time $T$ and extinction point $0$, the gradient profile behaves as $x\rightarrow0$ like $$\lim_{t\rightarrow T}\,(\nabla h)(x,t)\quad\sim\quad\frac{1}{\sqrt{2\beta}}\frac{x}{|x|}\frac{1}{\sqrt{|\log|x||}}\left[\frac{(\beta+1)^2}{8\beta}\frac{|x|^2}{|\log|x||}\right]^{\frac{1}{\beta+1}-\frac12},$$ agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions., Comment: 54 pages

In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a at blowup profile, already proved by Bricmont and Kupainen together with Herrero and Vel\'{a}zquez. Though our approach relies on the well celebrated method, based on the reduction of the problem to a finite dimensional one, then the use of a topological shooting method to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument., Comment: 31 pages

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., Comment: 61 pages

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha >0$. We study the nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe (note that since we consider fixed initial data, $\alpha$ and $\beta$ cannot be scaled away). But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions., Comment: 19 pages, 5 figures; v2: minor revision, matches accepted version

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1, 2], we constructed a solution $u$ for that equation such that $u$ and $\nabla u$ both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up., Comment: arXiv admin note: text overlap with arXiv:1911.06392

This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $\Omega$. While, for a given time $T>0$ and a point $a$ in this domain, we find a feedback control, which is acted on an internal subset $\omega$ of this domain, such that the corresponding solution to this system blows up at time $T$ and holds unique point $a$. We show that $a\in \omega$ can be the unique blowup point of the corresponding solution with a certain feedback control, and for any feedback control, $a\in \Omega\setminus \overline{\omega}$ could not be the unique blowup point., Comment: 40 pages

We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution which blows up in finite time $T$. Given a non isolated blow-up point $a$, we assume that the Taylor expansion of the solution near $(a,T)$ obeys some degenerate situation labeled by some even integer $m(a)\ge 4$. If we have a sequence $a_n \to a$ as $n\to \infty$, we show after a change of coordinates and the extraction of a subsequence that either $a_{n,1}-a_1 = o((a_{n,2}-a_2)^2)$ or $|a_{n,1}-a_1||a_{n,2}-a_2|^{-\beta} |\log|a_{n,2}-a_2||^{-\alpha} \to L> 0$ for some $L>0$, %up to extracting a subsequence still denoted the same, where $\alpha$ and $\beta$ enjoy a finite number of rational values with $\beta \in(0,2]$ and $L$ is a solution of a polynomial equation depending on the coefficients of the Taylor expansion of the solution. If $m(a)=4$, then $\alpha=0$ and either $\beta=3/2$ or $\beta =2$., Comment: 5 tables

We consider the semilinear heat equation $$\partial_t u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous., Comment: arXiv admin note: text overlap with arXiv:2012.05374

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, The result in one space dimension, has been proved in \cite{HZjmaa2020}. Our goal here is to extend this result to higher dimensions., Comment: arXiv admin note: substantial text overlap with arXiv:1906.12059

In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& \Delta u - u + \displaystyle{\frac{u^p}{ \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_\Omega u^r dr \right)^\gamma }}\quad\text{in}\quad \Omega \times (0,T), \\[0.2cm] \frac{ \partial u}{ \partial \nu} & = & 0 \text{ on } \Gamma = \partial \Omega \times (0,T),\\ u(0) & = & u_0, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. \cite{KSN17, NKMI2018}. Under the Turing type condition $$ \frac{r}{p-1} < \frac{N}{2}, \gamma r \ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $\Omega,$ i.e. $$ u(x_0, t) \sim (\theta^*)^{-\frac{1}{p-1}} \left[\kappa (T-t)^{-\frac{1}{p-1}} \right], $$ where $$ \theta^* := \lim_{t \to T} \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_\Omega u^r dr \right)^{- \gamma} \text{ and } \kappa = (p-1)^{-\frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) \sim ( \theta^* )^{-\frac{1}{p-1}} \left[ \frac{(p-1)^2}{8p} \frac{|x-x_0|^2}{ |\ln|x-x_0||} \right]^{ -\frac{1}{p-1}} \text{ as } x \to 0, $$ and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in \cite{MZnon97} and \cite{DZM3AS19}., Comment: 31 pages

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work (Abdelhedi-Zaag JDE 2021), we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.

We develop and study a time-space discrete discontinuous Galerkin finite elements method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several numerical examples and benchmarks., Comment: 34 pages, 8 figures, 3 tables

Crohn's disease is an inflammatory bowel disease (IBD) that is not well understood. In particular, unlike other IBDs, the inflamed parts of the intestine compromise deep layers of the tissue and are not continuous but separated and distributed through the whole gastrointestinal tract, displaying a patchy inflammatory pattern. In the present paper, we introduce a toy-model which might explain the appearance of such patterns. We consider a reaction-diffusion system involving bacteria and phagocyte and prove that, under certain conditions, this system might reproduce an activator-inhibitor dynamic leading to the occurrence of Turing-type instabilities. In other words, we prove the existence of stable stationary solutions that are spatially periodic and do not vanish in time. We also propose a set of parameters for which the system exhibits such phenomena and compare it with realistic parameters found in the literature. This is the first time, as far as we know, that a Turing pattern is investigated in inflammatory models.

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile., Comment: 85 pages. arXiv admin note: substantial text overlap with arXiv:1703.00081

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on the following method: we linearize the equation (in similarity variables) around the expected profile, then, we control the nonpositive directions of the spectrum thanks to the decreasing properties of the kernel. Finally, we use a topological argument to control the positive directions of the spectrum.

We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial results are given at the origin, where the degeneracy occurs. Some nontrivial obstacles, due to the nonconstant speed of propagation, have to be surmounted.

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, namely $u'' =|u|^{p-1}u\log^a (2+u^2)$ Unlike the pure power case ($g(u)=|u|^{p-1}u$) the difficulties here are due to the fact that equation (1) is not scale invariant., Comment: 37 pages

We construct a finite-time blow-up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension. Given any integer $k\ge 2$ and $\zeta_0 \in \mathbb{R}$, we construct a blow-up solution with a characteristic point $a$, such that the asymptotic behavior of the solution near $(a,T(a))$ shows a decoupled sum of $k$ solitons with alternate signs, whose centers (in the hyperbolic geometry) have $\zeta_0$ as a center of mass, for all times. Although the result is similar to the unperturbed case in its statement, our method is new. Indeed, our perturbed equation is not invariant under the Lorentz transform, and this requires new ideas. In fact, the main difficulty in this paper is to prescribe the center of mass $\zeta_0 \in \mathbb{R}$. We would like to mention that our method is valid also in the unperturbed case, and simplifies the original proof by C\^ote and Zaag \cite{CZcpam13}, as far as the center of mass prescription is concerned., Comment: 35 pages. arXiv admin note: text overlap with arXiv:1110.2512

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times \Omega:$ \begin{eqnarray*} \partial_t u = \Delta u +\displaystyle \frac{\lambda }{ (1-u)^2 \left( 1 +\displaystyle \gamma \int_{\Omega} \frac{1}{1-u} dx \right)^2}, \quad 0 \leq u <1, \end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $\lambda, \gamma > 0$. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point $a \in \Omega$. In particular, we give a description of the quenching behavior according to the following final profile $$ 1 - u(x,T) \sim \theta^*\left[ \frac{|x-a|^2}{|\ln|x-a||} \right]^\frac{1}{3} \text{ as } x \to a, \theta^* > 0.$$ The construction relies on some connection between the quenching phemonenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The proof relies on two main steps: A reduction to a finite dimensional problem and a topological argument based on Index theory. The highlight of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data., Comment: 56 pages

We consider in this note the semilinear heat system $$\partial_t u = \Delta u + f(v), \quad \partial_t v = \mu\Delta v + g(u), \quad \mu > 0,$$ where the nonlinearity has no gradient structure taking of the particular form $$f(v) = v|v|^{p-1} \quad \text{and}\quad g(u) = u|u|^{q-1} \quad \text{with} \quad p, q > 1, $$ or $$f(v) = e^{pv}\quad \text{and} \quad g(u) = e^{qu} \quad \text{with} \quad p,q > 0.$$ We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on two-step procedure: the reduction of the problem to a finite dimensional one via a spectral analysis, then solving the finite dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [Ann. IHP 2018] and [JDEs 2018].

We consider the higher-order semilinear parabolic equation $$ \partial_t u = -(-\Delta)^{m} u + u|u|^{p-1}, $$ in the whole space $\mathbb{R}^N$, where $p > 1$ and $m \geq 1$ is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by [Galaktionov, rspa2011], we revisit the technique developed by [Merle-Zaag, duke1997] for the classical case $m = 1$, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [Galaktionov, rspa2011].

We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + e^{pv}, \quad & \partial_t v = \mu \Delta v + e^{qu}, u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. \quad p, q, \mu > 0, $$ in the whole space $\mathbb{R}^N$. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem., Comment: 47 pages. add references, many typos have been corrected. arXiv admin note: text overlap with arXiv:1610.09883

We construct a solution for the Complex Ginzburg-Landau equation in some critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution., Comment: arXiv admin note: text overlap with arXiv:1506.08306

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 solutions., Comment: arXiv admin note: text overlap with arXiv:1306.4837

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = \Delta U + \alpha|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad \alpha > -1.$$ We construct for this equation a solution which blows up in finite time $T > 0$ and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time $T$ simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the change of the final blowup profile in comparison with the case $\alpha = 0$. The proof of the construction inspired by the method of Merle and Zaag in 1997, relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the \textit{blowup region}, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations of the initial data., Comment: 39 pages

We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + |v|^{p-1}v, \quad & \partial_t v = \mu \Delta v + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. $$ in the whole space $\mathbb{R}^N$, where $p, q > 1$ and $\mu > 0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0, v_0)$ simultaneously in $u$ and $v$ only at one blowup point $a$, according to the following asymptotic dynamics: $$\left\{\begin{array}{c} u(x,t)\sim \Gamma\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(p + 1)}{pq - 1}},\\ v(x,t)\sim \gamma\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(q + 1)}{pq - 1}}, \end{array}\right.$$ with $b = b(p,q,\mu) > 0$ and $(\Gamma, \gamma) = (\Gamma(p,q), \gamma(p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu = 1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data., Comment: 56 pages

We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with an isolated characteristic blow-up point at the origin, and a blow-up surface which is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in $R^2$. The blow-up surface is differentiable outside the bisecrtices. As for the asymptotic behavior in similariy variables, the solution converges to the classical one-dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two-dimensional stationary solution, whose existence is a by-product of the proof. At the origin, it behaves like the sum of 4 solitons localized on the two axes, with opposite signs for neighbors. This is the first example of a blow-up solution with a characteristic point in higher dimensions, showing a really two-dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of non-characteristic points where the blow-up surface is non-differentiable., Comment: 76 pages

We consider $u(x,t)$, a solution of $\partial_tu = \Delta u + |u|^{p-1}u$ which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to \mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to be the blow-up set of $u$, that is the set of all blow-up points. Under suitable nondegeneracy conditions, we show that if $S$ contains a $(N-\ell)$-dimensional continuum for some $\ell \in \{1,\dots, N-1\}$, then $S$ is in fact a $\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptotic behavior of $u$ near blow-up. In order to obtain such a refined behavior, we have to abandon the explicit profile function as a first order approximation and take a non-explicit function as a first order description of the singular behavior. This way we escape logarithmic scales of the variable $(T-t)$ and reach significant small terms in the polynomial order $(T-t)^\mu$ for some $\mu > 0$. The refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on $S$., Comment: 27 pages

We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. 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 uses the index theory to conclude. Thanks to the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and point, we also show the stability of the constructed solution with respect to initial data. This note presents the results and the main arguments. For the details, we refer to our paper \cite{TZ15}., Comment: Proceedings of the Conference EDP - Normnandie, Le Havre, October 21-22, 2015

We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times L^{\infty}$, we derive the blow-up rate near a non-characteristic point in the smaller space, and give some bounds near other points. Our result generalize those proved by Godin under high regularity assumptions on initial data.

We refine the asymptotic behavior of solutions to the semilinear heat equation with Sobolev subcritical power nonlinearity which blow up in some finite time at a blow-up point where the (supposed to be generic) profile holds. In order to obtain this refinement, we have to abandon the explicit profile function as a first order approximation, and take a non explicit function as a first order description of the singular behavior. This non explicit function is in fact a special solution which we construct, obeying some refined prescribed behavior. The construction relies on the reduction of the problem to a finite dimensional one and the use of a topological argument based on index theory to conclude. Surprisingly, the new non explicit profiles which we construct make a family with finite degrees of freedom, namely $\frac{(N+1)N}{2}$ if $N$ is the dimension of the space., Comment: 42 pages

We consider the nonlinear heat equation with a nonlinear gradient term: $\partial_t u =\Delta u+\mu|\nabla u|^q+|u|^{p-1}u,\; \mu>0,\; q=2p/(p+1),\; p>3,\; t\in (0,T),\; x\in \R^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t)^{1/2}|\log(T-t)|^{1/2},$ like in the standard nonlinear heat equation, i.e. $\mu=0,$ but as $(T-t)^{1/2}|\log(T-t)|^{\beta}$ with $\beta=(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $\nabla u$ blow up simultaneously and at a single point, and give the final profile. In particular, the final profile is more singular than the case of the standard nonlinear heat equation.

We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, 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. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data., Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:1306.4435