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Razvan Gabriel Iagar

Publications and source records attributed to Razvan Gabriel Iagar.

At least 55 records · Page 3Linked to original sources
arXiv2022

Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$, $q\in(0,1)$ and $σ>0$, is addressed. Introducing the critical exponent $σ^* := 2(1-q)/(m-1)$ for $m>1$ and $σ_*=\infty$ for $m=1$, extinction in finite time is known to take place for $σ\in [0,σ^*)$ and an alternative proof is provided therein. When $m>1$ and $σ\ge σ^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of $σ$ and showing their sharpness.

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arXiv2022

A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential

Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=Δu^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p 0$ and $p\in[1,\infty)$. As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition $u_0$, contrasting with previous results in literature for the critical limit $p=m$.

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arXiv2022

Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=Δu^m-|x|^σu^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $σ> 2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 \in L^\infty(\mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t > 0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.

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arXiv2022

Non-existence of nonnegative separate variable solutions to a porous medium equation with spatially dependent nonlinear source

The non-existence of nonnegative compactly supported classical solutions to $$- ΔV(x) - |x|^σV(x) + \frac{V^{1/m}(x)}{m-1} = 0, \qquad x\in\mathbb{R}^N,$$ with $m>1$, $σ>0$, and $N\ge 1$, is proven for $σ$ sufficiently large. More precisely, in dimension $N\geq4$, the optimal lower bound on $σ$ for non-existence is identified, namely $$σ\geqσ_c := \frac{2(m-1)(N-1)}{3m+1},$$ while, in dimensions $N\in\{1,2,3\}$, the lower bound derived on $σ$ improves previous ones already established in the literature. A by-product of this result is the non-existence of nonnegative compactly supported separate variable solutions to a porous equation medium equation with spatially dependent superlinear source.

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arXiv2021

Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential

We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=Δu^m+|x|^σu^p, $$ posed in dimension $N\geq2$, with $m>1$, $σ\in(-2,0)$ and $1 1$ and $p>1$, showing an interesting effect induced by the singular potential $|x|^σ$. This result is also applied to reaction-diffusion equations with general potentials $V(x)$ to prevent finite time blow-up via comparison.

math.AP↗
arXiv2021

Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents $m>1$, $p\in(0,1)$ and $σ>2(1-p)/(m-1)$. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are \emph{compactly supported} and might present two different types of interface behavior and three different possible \emph{good behaviors} near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of $σ$. This paper generalizes in dimension $N>1$ previous results by the authors in dimension $N=1$ and also includes some finer classification of the profiles for $σ$ large that is new even in dimension $N=1$.

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arXiv2021

Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction

We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in $\real^N$ with $N\geq3$, where $$ 0 1, $$ and the critical value for the weight $$ σ=\frac{2(p-1)}{1-m}. $$ The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a \emph{change of sign of both self-similar exponents} at $m=m_s=(N-2)/(N+2)$, leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a \emph{perfect equilibrium} in the competition between the fast diffusion and the reaction effects.

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arXiv2021

Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=Δu^m + |x|^σu^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $σ>0$. A new and explicit critical exponent $$ σ_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $σ_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$.

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arXiv2021

Eternal solutions for a reaction-diffusion equation with weighted reaction

We prove existence and uniqueness of \emph{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in $\real^N$, with $m>1$, $0 2$. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.

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arXiv2020

Self-similar blow-up profiles for a reaction-diffusion equation with critically strong weighted reaction

We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0 2$ completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for $σ>2$. Moreover all the blow-up profiles have compact support and their supports are \emph{localized}: there exists an explicit $η>0$ such that any blow-up profile satisfies ${\rm supp}\,f\subseteq[0,η]$. This property is unexpected and contrasting with the range $m+p>2$. We also classify the possible behaviors of the profiles near the origin.

math.AP↗
arXiv2020

Self-similar blow-up profiles for a reaction-diffusion equation with strong weighted reaction

We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0 2(1-p)/(m-1)$. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for $m+p>2$ and $σ$ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when $p<1$. We moreover prove that, if the condition $m+p>2$ is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist \emph{two different interface behaviors} for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases \emph{global blow-up} occurs, and in some other cases finite time blow-up occurs \emph{only at space infinity}. We also show that there is no self-similar solution if $m+p<2$, while the critical range $m+p=2$ with $σ>2$ is postponed to a different work due to significant technical differences.

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arXiv2019

Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations

In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation $$ \partial_t u=(u^m)_{xx}+a(x)(u^m)_x+b(x)u^m, $$ posed for $x\in\real$, $t\geq0$ and $m>1$, where $a$, $b$ are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type $$ f(y)\partial_τθ=(θ^m)_{yy}, $$ posed in the half-line $y\in[0,\infty)$ with $τ\geq0$, $m>1$ and suitable density functions $f(y)$. We apply this correspondence to the case of constant coefficients $a(x)=1$ and $b(x)=K>0$. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as $x\to-\infty$. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when $f(y)=y^{-γ}$ with $γ>2$.

math.AP↗
arXiv2019

Blow up profiles for a reaction-diffusion equation with critical weighted reaction

We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^σu^m, $$ posed for $(x,t)\in\real\times(0,T)$, with $m>1$ and $σ>0$. In strong contrast with the well-studied equation without the weight (that is $σ=0$), on the one hand we show that for $σ>0$ sufficiently small there exist \emph{multiple self-similar profiles with interface} at a finite point, more precisely, given any positive integer $k$, there exists $δ_k>0$ such that for $σ\in(0,δ_k)$, there are at least $k$ different blow up profiles with compact support and interface at a positive point. On the other hand, we also show that for $σ$ sufficiently large, the blow up self-similar profiles with interface \emph{cease to exist}. This unexpected balance between existence of multiple solutions and non-existence of any, when $σ>0$ increases, is due to the effect of the presence of the weight $|x|^σ$, whose influence is the main goal of our study. We also show that for any $σ>0$, there are no blow up profiles supported in the whole space, that is with $u(x,t)>0$ for any $x\in\real$ and $t\in(0,T)$.

math.AP↗
arXiv2019

Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction

We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ in the range of exponents $1 0$. We classify blow up solutions in self-similar form, that are likely to represent typical blow up patterns for general solutions. We thus show that the non-homogeneous coefficient $|x|^σ$ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for $σ\sim0$, blow up profiles have similar behavior to the well-established profiles for the homogeneous case $σ=0$, and typically \emph{global blow up} occurs, while for $σ>0$ sufficiently large, there exist blow up profiles for which blow up \emph{occurs only at space infinity}, in strong contrast with the homogeneous case. This work is a part of a larger program of understanding the influence of unbounded weights on the blow up behavior for reaction-diffusion equations.

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arXiv2016

Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption $\partial_t u - Δ_p u + |\nabla u|^{p-1} = 0$ in $(0, \infty) \times \mathbb{R}^N$, and fast diffusion $2N/(N + 1) < p < 2$. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as $|x| \to \infty$, it is shown that the corresponding solution $u$ to the above equation approaches a uniquely determined separate variable solution of the form $U(t, x) = (T_e - t)^{1/(2-p)} f_* (|x|), (t, x) \in (0, T_e) \times \mathbb{R}^N$, as $t \to T_e$, where $T_e$ denotes the finite extinction time of $u$. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile $f_*$ is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional.

math.AP↗
arXiv2015

Asymptotic behavior for the critical nonhomogeneous porous medium equation in low dimensions

We deal with the large time behavior for a porous medium equation posed in nonhomogeneous media with singular critical density $$ |x|^{-2}\partial_tu(x,t)=Δu^m(x,t), \quad (x,t)\in \real^N\times(0,\infty), \ m\geq1, $$ posed in dimensions $N=1$ and $N=2$, which are also interesting in applied models according to works by Kamin and Rosenau. We deal with the Cauchy problem with bounded and continuous initial data $u_0$. We show that in dimension $N=2$, the asymptotic profiles are self-similar solutions that vary depending on whether $u_0(0)=0$ or $u_0(0)=K\in(0,\infty)$. In dimension $N=1$, things are strikingly different, and we find new asymptotic profiles of an unusual mixture between self-similar and traveling wave forms. We thus complete the study performed in previous recent works for the bigger dimensions $N\geq3$.

math.AP↗
arXiv2015

Instantaneous shrinking and single point extinction for viscous Hamilton-Jacobi equations with fast diffusion

For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation $\partial\_t u-Δ\_p u+|\nabla u|^q=0$ in $(0,\infty)\times\real^N$ are known to vanish identically after a finite time when $2N/(N+1) \textless{} p \leq 2$ and $q\in(0,p-1)$. Further properties of this extinction phenomenon are established herein: \emph{instantaneous shrinking} of the support is shown to take place if the initial condition $u\_0$ decays sufficiently rapidly as $|x|\to\infty$, that is, for each $t \textgreater{} 0$, the positivity set of $u(t)$ is a bounded subset of $\real^N$ even if $u\_0 \textgreater{} 0$ in $\real^N$. This decay condition on $u\_0$ is also shown to be optimal by proving that the positivity set of any solution emanating from a positive initial condition decaying at a slower rate as $|x|\to\infty$ is the whole $\real^N$ for all times. The time evolution of the positivity set is also studied: on the one hand, it is included in a fixed ball for all times if it is initially bounded (\emph{localization}). On the other hand, it converges to a single point at the extinction time for a class of radially symmetric initial data, a phenomenon referred to as \emph{single point extinction}. This behavior is in sharp contrast with what happens when $q$ ranges in $[p-1,p/2)$ and $p\in (2N/(N+1),2]$ for which we show \emph{complete extinction}. Instantaneous shrinking and single point extinction take place in particular for the semilinear viscous Hamilton-Jacobi equation when $p=2$ and $q\in (0,1)$ and seem to have remained unnoticed.

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