arXiv subjects

Slim Tayachi

Publications and source records attributed to Slim Tayachi.

18 recordsLinked to original sources

arXiv2026

Threshold for the existence of scattering states for inhomogeneous nonlinear Schr\"odinger equations without gauge invariance

We consider the asymptotic behavior of solutions to inhomogeneous nonlinear Schr\"odinger equations with non-gauge-invariant nonlinearities. The spatial coefficient in the nonlinear term may have different orders of singularity at the origin and decay at infinity. Under a coercivity condition involving the coefficient and the nonlinearity, we show that no scattering states exist for the equation below a Strauss-type threshold determined by the decay at infinity. Our class includes nonlinearities with a dominant non-oscillatory component. We also prove a complementary small-data scattering result above this threshold, using non-admissible Strichartz estimates in Lorentz spaces adapted to the singular coefficient.

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arXiv2025

Asymptotically self-similar global solutions for Hardy-H\'enon parabolic equations

We construct asymptotically self-similar global solutions to the Hardy-H\'enon parabolic equation $\partial_t u - \Delta u = \pm |x|^{\gamma} |u|^{\alpha-1} u$, $\alpha>1$, $\gamma \in \mathbb{R}$ for a large class of initial data belonging to weighted Lorentz spaces. The solution may be asymptotic to a self-similar solution of the linear heat equation or to a self-similar solution to the Hardy-H\'enon parabolic equation depending on the speed of decay of the initial data at infinity. The asymptotic results are new for the H\'enon case $\gamma>0$. We also prove the stability of the asymptotic profiles. Our approach applies for $\gamma> -\min(2,d)$ and unifies the cases $\gamma>0$, $\gamma=0$ and $-\min(2,d)<\gamma<0$. For complex-valued initial data, a more intricate asymptotic behaviors can be shown; if either one of the real part or the imaginary part of the initial data has a faster spatial decay, then the solution exhibits a combined Nonlinear-"Modified Linear" asymptotic behavior, which is completely new even for the Fujita case $\gamma=0$. In Appendix, we show the non-existence of local positive solutions for supercritical initial data.

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arXiv2023

Asymptotic behavior and life-span estimates for the damped inhomogeneous nonlinear Schrödinger equation

We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schrödinger equation $ i\partial_tu+Δu+μ|x|^{-b}|u|^αu+iau=0$, $μ\in\mathbb{C} $, $b>0$, $a \in \mathbb{C}$ such that $\Re \textit{e}(a) \geq 0$, $α>0$. We establish lower and upper bound estimates of the life-span. In particular for $a\geq 0$, we obtain explicit values $a_*,\; a^*$ such that if $a a^*,$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$

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arXiv2023

Unconditional uniqueness and non-uniqueness for Hardy-H\'enon parabolic equations

We study the problems of uniqueness for Hardy-H\'enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H\'enon type) in the nonlinear term. To deal with the Hardy-H\'enon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-H\'enon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.

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arXiv2022

New life-span results for the nonlinear heat equation

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation $\partial_tu-Δu=|u|^αu,\;α>0$ with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of life-span are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for life-span using dilation methods and we give new life-span estimates for Hardy-Hénon parabolic equations.

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arXiv2022

Existence and regularity of source-type self-similar solutions for stable thin-film equations

We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h_t=-(h^nh_{zzz})_z+(h^{n+3})_{zz},$ $ t>0,\; z\in \mathbb{R};\; h(0,z)= ωδ(z)$ where $n\in (\frac{3}{2},3),\; ω> 0$ and $δ$ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a traveling-wave solution for the standard thin-film equation: $h_t=-(h^nh_{zzz})_z$. In this paper we sharpen this result, proving that the higher order corrections are analytic with respect to three variables: the first one is just the {spatial} variable, whereas the second and the third (except for $n = 2$) are irrational powers of it. It is known that this third variable does not appear for the thin-film equation without gravity.

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arXiv2021

Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

In this paper we consider the inhomogeneous nonlinear Schrödinger equation $i\partial_t u +Δu=K(x)|u|^αu,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$ $0<b<\min(2,N-2s),$ $0<α<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $α_0(b)<α<(4-2b)/N$. The value $α_0(b)$ is the positive root of $Nα^2+(N-2+2b)α-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=μ|x|^{-b}$, $μ\in \mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $α$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.

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arXiv2020

Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting

We investigate the large time behavior of compactly supported solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

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arXiv2019

Large time behavior of solutions to the nonlinear heat equation with absorption with highly singular antisymmetric initial values

In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $ u_t - Δu + |u|^αu =0$, where $u=u(t,x)\in {\mathbb R}, $ $(t,x)\in (0,\infty)\times{\mathbb R}^N$ and $α>0$. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables $x_1,\; x_2,\; \cdots,\; x_m$ for some $m\in \{1,2, \cdots, N\}$, such as $u_0 = (-1)^m\partial_1\partial_2 \cdots \partial_m|\cdot|^{-γ} \in {{\mathcal S'}({\mathbb R}^N)}$, $0 < γ< N$. In fact, we show global well-posedness for initial data bounded in an appropriate sense by $u_0$, for any $α>0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $Ω_m = \{x \in {{\mathbb R}^N} : x_1, \cdots, x_m > 0\}$, and then to extend the results by reflection to solutions on ${{\mathbb R}^N}$ which are antisymmetric. We show that the large time behavior depends on the relationship between $α$ and $2/(γ+m)$, and we consider all three cases, $α$ equal to, greater than, and less than $2/(γ+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

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arXiv2019

Global existence and decay estimates for the heat equation with exponential nonlinearity

In this paper we consider the initial value {problem $\partial_{t} u- Δu=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f : \mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|\sim \mbox{e}^{|u|^q}$ as $|u|\to \infty$,} $|f(u)|\sim |u|^{m}$ as $u\to 0,$ $0 1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$

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arXiv2018

Well-posedness, Global existence and decay estimates for the heat equation with general power-exponential nonlinearities

In this paper we consider the problem: $\partial_{t} u- Δu=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$ for $f(u)\sim \mbox{e}^{|u|^q},\;0 0,$ $\displaystyle\liminf_{s\to \infty}\left(f(s)\,{\rm{e}}^{-λs^p}\right)>0,$ then non-existence occurs in $\exp L^p(\R^N).$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\sim |u|^{m}$ as $u\to 0,$ ${N(m-1)\over 2}\geq p$, we show that the solution is global. In particular, $p-1>0$ sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m$.

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arXiv2017

The nonlinear heat equation involving highly singular initial values and new blowup and life span results

In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = Δu +a |u|^αu, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; α>0;$ with initial value $u(0)\in L^1_{\rm{loc}}\left({\mathbb R}^N\setminus\{0\}\right)$, anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\; x_m$ and $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-γ})$ for $x_1>0,\; \cdots,\; x_m>0,$ where $C>0$ is a constant, $m\in \{1,\; 2,\; \cdots,\; N\},$ $0<γ 0$ is sufficiently small. We also construct blowing up solutions with initial data $λ_n f$ such that $λ_n^{[({1\over α}-{γ+m\over 2})^{-1}]}T_{\max}(λ_n f)$ has different finite limits along different sequences $λ_n\to 0$. Our result extends the known "small lambda" blow up results for new values of $α$ and a new class of initial data.

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arXiv2017

Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity

In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ Δ^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.

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arXiv2016

Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term

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}.

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arXiv2015

Single-point blow-up for parabolic systems with exponential nonlinearities and unequal diffusivities

We study positive blowing-up solutions of systems of the form: $$u_t=δ_1 Δu+e^{pv},\quad v_t= δ_2Δv+e^{qu},$$ with $δ_1,δ_2>0$ and $p, q>0$. We prove single-point blow-up for large classes of radially decreasing solutions. This answers a question left open in a paper of Friedman and Giga~(1987), where the result was obtained only for the equidiffusive case $δ_1=δ_2$ and the proof depended crucially on this assumption.

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arXiv2015

Existence of a Stable Blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term

We consider the nonlinear heat equation with a nonlinear gradient term: $\partial_t u =Δu+μ|\nabla u|^q+|u|^{p-1}u,\; μ>0,\; q=2p/(p+1),\; p>3,\; t\in (0,T),\; x\in \R^N.$ 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. $μ=0,$ but as $(T-t)^{1/2}|\log(T-t)|^β$ with $β=(p+1)/[2(p-1)]>1/2.$ 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.

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arXiv2015

Improved conditions for single-point blow-up in reaction-diffusion systems

We study positive blowing-up solutions of the system: $$u_{t}-δΔu=v^p,\,\,\, v_{t}-Δv=u^{q},$$ as well as of some more general systems. For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions: (i) no type I blow-up assumption is made (and it is known that this property may fail); (ii) no equidiffusivity is assumed, i.e. any $δ>0$ is allowed; (iii) a large class of nonlinearities $F(u,v)$, $G(u,v)$ can be handled, which need not follow a precise power behavior. As side result, we also obtain lower pointwise estimates for the final blow-up profiles.

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arXiv2013

Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation

We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: $u_t=D_x^{2α} u \mp u^2,\; t\in (0,T),\; x\in \R$ or $ \T $, with $ 0<α\le 1 $ is well-posed in $ H^s $ for $ s\ge \max(-α,1/2-2α) $ except in the case $ α=1/2 $ where it is shown to be well-posed for $ s>-1/2 $ and ill-posed for $ s=-1/2 $. As a by-product we improve the known well-posedness results for the heat equation ($α=1$) by reaching the end-point Sobolev index $ s=-1 $. Finally, in the case $ 1/2<α\le 1 $, we also prove optimal results in the Besov spaces $B^{s,q}_2.$

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