arXiv subjects

Philippe Souplet

Publications and source records attributed to Philippe Souplet.

At least 37 records · Page 2Linked to original sources

arXiv2021

Complete classification of gradient blow-up and recovery of boundary condition for the viscous Hamilton-Jacobi equation

We study the Cauchy-Dirichlet pbm for superquadratic viscous Hamilton-Jacobi eq. We give a complete classification, namely rates and space-time profiles, in 1d case when viscosity sol. undergo gradient blow-up (GBU) or recovery of boundary condition (RBC) at any time when such phenomenon occurs. These results can be modified in radial domains in general dimensions. Previously, upper and lower estimates of GBU or RBC rates were available only in special case when basic comparison principle can be used. Even for type II BU in other PDEs, as far as we know, there has been no complete classification except [50], in which the argument relies on features peculiar to chemotaxis syst. Whereas there are many results on construction of special type II BU sol. of PDEs with investigation of (in-)stability of bubble, determination of (in-)stability of space-time profile for general sol. has not been done. In this paper, we determine whether space-time profile for each sol. is stable or unstable. A key in our proofs is to focus on algebraic structure with respect to vanishing intersections with singular steady state. In turn, GBU and RBC rates and profiles, as well as their (in-)stability, can be completely characterized by the number of vanishing intersections. We construct special sol. in bounded and unbounded intervals in both GBU and RBC cases, based on methods from [29], and then apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space-time profile. In the process, careful construction of special sol. with specific behaviors in intermediate and outer regions, far from bubble and the RBC point, plays essential role. The application of such techniques to viscosity sol. is completely new.

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arXiv2021

On refined blowup estimates for the exponential reaction-diffusion equation

We consider radial decreasing solutions of the semilinear heat equation with exponential nonlinearity. We provide a relatively simple proof of the sharp upper estimates for the final blowup profile and for the refined space-time behavior. We actually establish a global, upper space-time estimate, which contains those of the final and refined profiles as special cases.

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arXiv2021

The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients

We develop a new, unified approach to the following two classical questions on elliptic PDE: the strong maximum principle for equations with non-Lipschitz nonlinearities, and the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

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arXiv2021

Singularity formation and regularization at multiple times in the viscous Hamilton-Jacobi equation

The Cauchy-Dirichlet pbm for the superquadratic viscous Hamilton-Jacobi eqn (VHJ), which has important applications in stochastic control theory, admits a unique, global viscosity solution. Sol. thus exist in the weak sense after appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. Whereas theory of visc. sol. has been extensively studied and applied to many PDEs, there are less results on refined behavior of sol. In particular, detailed behavior of visc. sol. of VHJ after GBU has remained mostly open. Here, in general dim., for each $m\ge 1$ we construct sol. which undergo GBU and LBC at least at $m$ times and then recover regularity, as well as sol. that exhibit GBU without LBC at 1st blowup time. In 1d, we obtain the complete classification of visc. sol. at each time, which extends to radial cases in higher d. Furthermore for each $m\ge 2$ and arbitrarily given combination of GBU types with/without LBC at $m$ times in arbitrarily given order, we show exist. of a sol. with this exact combination of GBU. Some sol. display a new type of behavior called "bouncing". Global weak sol. of VHJ with multiple time singularity turn out to display larger variety of behaviors than for the Fujita eqn. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires a delicate argument. Since we do not rely on any known special sol. unlike in Fujita eqn, our method is expected to apply to other eqns. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. In this framework our results imply that for certain spatial distributions of rewards, if a controled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.

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arXiv2020

A Liouville-type theorem for the Lane-Emden equation in a half-space

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.

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arXiv2020

Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.

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arXiv2019

Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton-Jacobi equations

Consider the diffusive Hamilton-Jacobi equation $$u_t-Δu=|\nabla u|^p+h(x)\ \ \text{ in } Ω\times(0,T)$$ with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with $p>2$. We first consider the case of time-increasing solutions. For such solutions, the precise rate was obtained by Guo and Hu (2008) in one space dimension, but the higher dimensional case has remained an open question (except for radially symmetric solutions in a ball). Here, we partially answer this question by establishing the optimal estimate $$C_1(T-t)^{-1/(p-2)}\leq \|\nabla u(t)\|_{\infty} \leq C_2(T-t)^{-1/(p-2)} \tag{1}$$ for time-increasing gradient blowup solutions in any convex, smooth bounded domain $Ω$ with $2 2$, we show that more singular rates may occur for solutions which are $\textit{not}$ time-increasing. Namely, for a suitable class of solutions in one space-dimension, we prove the lower estimate $\|u_x(t)\|_\infty \geq C(T-t)^{-2/(p-2)}$.

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arXiv2019

Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation

We shed light on a long-standing open question for the semilinear heat equation $u_t = Δu + |u|^{p-1} u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$Ω$, we prove that the critical $L^q$ norm blows up whenever the solution undergoes {\it type~I~blow-up.} A~similar property is also obtained for the local critical $L^q$ norm near any blow-up point. In view of recent results of existence of type~II blow-up solutions with bounded critical $L^q$ norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical $L^q$ norm blow-up appears to be a completely new observation. Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with {\it backward uniqueness and unique continuation properties} for parabolic equations. As a by-product, we obtain the nonexistence of self-similar profiles in the critical $L^q$ space. Such properties were up to now only known for $ p \le p_S $ and in radially symmetric case for $ p > p_S $, where $ p_S $ is the Sobolev exponent.

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arXiv2019

A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

We consider the elliptic equation $-Δu = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given.

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arXiv2019

Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

We consider the nonlinear heat equation $u_t-Δu =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive solutions and obtain an optimal Fujita-type result: any positive solution blows up in finite time if $p\leq 1+\frac{2}{n}$ or $q\leq 1+\frac{1}{n+1}$, while global classical positive solutions exist for suitably small initial data when $p>1+\frac{2}{n}$ and $q> 1+\frac{1}{n+1}$. Although finite time blow-up cannot be produced by the gradient term alone and should be considered as an effect of the source term $|u|^p$, this result shows that the gradient term induces an interesting phenomenon of discontinuity of the critical Fujita exponent, jumping from $p=1+\frac{2}{n}$ to $p=\infty$ as $q$ reaches the value $1+\frac{1}{n+1}$ from above. Next, we investigate the case of sign-changing solutions and show that if $p\le 1+\frac{2}{n}$ or $0<(q-1)(np-1)\le 1$, then the solution blows up in finite time for any nontrivial initial data with nonnegative mean. Finally, a Fujita-type result, with a different critical exponent, is % also obtained for sign-changing solutions to the inhomogeneous version of this problem.

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arXiv2019

A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations with homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) for the parabolic problem in general bounded domains. Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior, with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions {\it generically} lose boundary conditions after GBU. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.

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arXiv2018

Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation

We study the qualitative properties of the unique global viscosity solution of the superquadratic diffusive Hamilton-Jacobi equation with (generalized) homogeneous Dirichlet conditions. We are interested in the phenomena of gradient blow-up (GBU), loss of boundary conditions (LBC), recovery of boundary conditions and eventual regularization, and in their mutual connections. In any space dimension, we establish the sharp minimal rate of GBU. Only partial results were previously known except in one space dimension. We also obtain the corresponding minimal regularization rate. In one space dimension, under suitable conditions on the initial data, we give a quite detailed description of the behavior of solutions for all $t>0$. In particular, we show that nonminimal GBU solutions immediately lose the boundary conditions after the blow-up time and are immediately regularized after recovering the boundary data. Moreover, both GBU and regularization occur with the minimal rates, while loss and recovery of boundary data occur with linear rate. We describe further the intermediate singular life of those solutions in the time interval between GBU and regularization. We also study minimal GBU solutions, for which GBU occurs {\it without} LBC: those solutions are immediately regularized, but their GBU and regularization rates are more singular. Most of our one-dimensional results crucially depend on zero-number arguments, which do not seem to have been used so far in the context of viscosity solutions of Hamilton-Jacobi equations.

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arXiv2018

Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth

We consider the Neumann and Cauchy problems for positivity preserving reaction-diffusion systems of $m$ equations enjoying the mass and entropy dissipation properties. We show global classical existence in any space dimension, under the assumption that the nonlinearities have at most quadratic growth. This extends previously known results which, in dimensions $n\ge 3$, required mass conservation and were restricted to the Cauchy problem. Our proof is also simpler.

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arXiv2018

Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for $n\ge 3$, we show that the final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with convergence in $L^1$ as $t\to T$. This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space-time estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not exist in dimension $2$. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.

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arXiv2017

Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity

We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, based on a parabolic equation with singular nonlinearity. We study the touchdown or quenching phenomenon. Recently, the question whether or not touchdown can occur at zero points of the permittivity profile, which had long remained open, was answered negatively by Guo and Souplet for the case of interior points, and we then showed that touchdown can actually be ruled out in subregions of the domain where the permittivity is positive but suitably small. The goal of this paper is to further investigate the touchdown localization problem and to show that, in one space dimension, one can obtain quite quantitative conditions. Namely, for large classes of typical, one-bump and two-bump permittivity profiles, we find good lower estimates of the ratio between f and its maximum, below which no touchdown occurs outside of the bumps. The ratio is rigorously obtained as the solution of a suitable finite-dimensional optimization problem (with either three or four parameters), which is then numerically estimated. Rather surprisingly, it turns out that the values of the ratio are not "small" but actually up to the order 0.3, which could hence be quite appropriate for robust use in practical MEMS design. The main tool for the reduction to the finite-dimensional optimization problem is a quantitative type I, temporal touchdown estimate. The latter is proved by maximum principle arguments, applied to a multi-parameter family of refined, nonlinear auxiliary functions with cut-off.

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arXiv2017

No touchdown at points of small permittivity and nontrivial touchdown sets for the MEMS problem

We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, involving a parabolic equation with singular nonlinearity. We study the touchdown, or quenching, phenomenon. Recently, the question whether or not touchdown can occur at zero points of the premittivity profile f, which had long remained open, was answered negatively for the case of interior points. The first aim of this article is to go further by considering the same question at points of positive but small permittivity. We show that, in any bounded domain, touchdown cannot occur at an interior point where the permittivity profile is suitably small. We also obtain a similar result in the boundary case, under a smallness assumption on f in a neighborhood of the boundary. This allows in particular to construct f producing touchdown sets concentrated near any given sphere. Our next aim is to obtain more information on the structure and properties of the touchdown set. In particular, we show that the touchdown set need not in general be localized near the maximum points of the premittivity profile f. In the radial case in a ball, we show the existence of M-shaped profiles for which the touchdown set is located far away from the maximum points of f and we even obtain strictly convex f for which touchdown occurs at the unique minimum point of f. These results show that some kind of smallness condition as above cannot be avoided in order to rule out touchdown at a point. On the other hand, we construct profiles f producing more complex behaviors: in any bounded domain the touchdown set may be concentrated near two arbitrarily given points, or two arbitrarily given (n-1)-dimensional spheres in a ball. These examples are obtained as a consequence of stability results for the touchdown time and touchdown set under small perturbations of the permittivity profile.

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arXiv2016

Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation

We consider the diffusive Hamilton-Jacobi equation, with homogeneous Dirichlet conditions and regular initial data. It is known from [Barles-DaLio, 2004] that the problem admits a unique, continuous, global viscosity solution, which extends the classical solution in case gradient blowup occurs. We study the question of the possible loss of boundary conditions after gradient blowup, which seems to have remained an open problem till now. Somewhat surprisingly, our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena. For any smooth bounded domain, we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary, as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup. Actually, we show that the latter possibility is rather exceptional. More generally, we show that the set of the points where boundary conditions are lost, can be prescribed to be arbitrarily close to any given open subset of the boundary.

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arXiv2016

The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation

We consider the diffusive Hamilton-Jacobi equation $$u_t-Δu=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the precise final asymptotic profile, under the form $$u_y(x,y,T) \sim d_p\Bigl[y+C|x|^{2(p-1)/(p-2)}\Bigr]^{-1/(p-1)},\quad\hbox{as $(x,y)\to (0,0)$.}$$ Interestingly, this result displays a new phenomenon of strong anisotropy of the profile, quite different to what is observed in other blowup problems for nonlinear parabolic equations, with the exponents $1/(p-1)$ in the normal direction $y$ and $2/(p-2)$ in the tangential direction $x$. Furthermore, the tangential profile violates the (self-similar) scale invariance of the equation, whereas the normal profile remains self-similar.

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