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Razvan Gabriel Iagar

Publications and source records attributed to Razvan Gabriel Iagar.

At least 37 records · Page 2Linked to original sources
arXiv2025

A new pair of transformations and applications to generalized informational inequalities and Hausdorff moment problem

We introduce a pair of transformations, which are mutually inverse, acting on rather general classes of probability densities in R. These transformations have the property of interchanging the main informational measures such as p-moments, Shannon and Rényi entropies, and Fisher information. We thus apply them in order to establish extensions and generalizations of the Stam and moment-entropy inequalities in a mirrored domain of the entropic indexes. Moreover, with the aid of the two transformations we establish formal solutions to the Hausdorff entropic moment problem by connecting them with the solutions of the standard Hausdorff problem. In addition, we introduce a Fisher-like moment problem and relate it to the standard Hausdorff moment problem.

math-ph↗
arXiv2025

A parabolic Hardy-Hénon equation with quasilinear degenerate diffusion

Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-Hénon equation involving a quasilinear degenerate diffusion and a space-dependent superlinear source featuring a singular potential $$\partial_t u=Δu^m+|x|^σu^p,\quad t>0,\ x\in\mathbb{R}^N,$$ when $m>1$, $p>1$ and $σ\in \big(\max\{-2,-N\},0 \big)$. While the superlinear source induces finite time blow-up when $σ=0$, whatever the value of $p>1$, at least for sufficiently large initial conditions, a striking effect of the singular potential $|x|^σ$ is the prevention of finite time blow-up for suitably small values of $p$, namely, $1 p_G$, is obtained by employing the Caffarelli-Kohn-Nirenberg inequalities. Another interesting feature is that uniqueness and comparison principle hold true for generic non-negative initial conditions when $p>p_G$, but their validity is restricted to initial conditions which are positive in a neighborhood of $x=0$ when $p\in (1,p_G)$, a range in which non-uniqueness holds true without this positivity condition. Finite time blow-up of any non-trivial, non-negative solution is established when $p_G p_F$. Optimal temporal growth rates are also derived for global solutions when $p\in (1,p_G]$. All the results are sharp with respect to the exponents $(m,p,σ)$ and conditions on $u_0$.

math.AP↗
arXiv2025

Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty) $$ with $$ m_c=\frac{(N-2)_+}{N}\leq m<1, \quad σ\in(\max\{-2,-N\},\infty), \quad p>\max\left\{1+\frac{σ(1-m)}{2},1\right\} $$ are considered. It is proved that global self-similar solutions with the specific tail behavior $$ u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty $$ exist exactly for $p\in(p_F(σ),p_s(σ))$, where $$ p_F(σ)=m+\frac{σ+2}{N}, \qquad p_s(σ)=\left\{\begin{array}{ll}\frac{m(N+2σ+2)}{N-2}, & N\geq3,\\\infty, & N\in\{1,2\}, \end{array}\right. $$ are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any $σ\in(-2,0)$ and $p$ as above, but do not exist for any $σ\geq0$ and $p\in(p_F(σ),p_s(σ))$. We stress that all these results are \emph{new also in the homogeneous case $σ=0$}.

math.AP↗
arXiv2024

Qualitative properties of solutions to a generalized Fisher-KPP equation

The following Fisher-KPP type equation $$ u_t=Ku_{xx}-Bu^q+Au^p, \quad (x,t)\in\real\times(0,\infty), $$ with $p>q>0$ and $A$, $B$, $K$ positive coefficients, is considered. For both $p>q>1$ and $p>1$, $q=1$, we construct stationary solutions, establish their behavior as $|x|\to\infty$ and prove that they are separatrices between solutions decreasing to zero in infinite time and solutions presenting blow-up in finite time. We also establish decay rates for the solutions that decay to zero as $t\to\infty$.

math.AP↗
arXiv2024

A critical non-homogeneous heat equation with weighted source

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=Δu+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are obtained, in the range of exponents $p>1$, $σ\ge-2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as $t\to\infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case $σ=-2$ we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher-KPP equation is derived and employed in order to deduce these properties.

math.AP↗
arXiv2024

A Hardy-H{é}non equation in $\mathbb{R}^N$ with sublinear absorption

Consider $m\>1$, $N\ge 1$ and $\max\{-2,-N\}\<σ\<0$. The Hardy-Hénon equation with sublinear absorption\begin(equation*}- Δv(x) - |x|^σv(x) + \frac{1}{m-1} v^{1/m}(x)= 0, \qquad x\in\mathbb{R}^N,\end{equation*}is shown to have at least one solution $v\in H^1(\mathbb{R}^N)\cap L^{(m+1)/m}(\mathbb{R}^N)$, which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity $v\in W^{2,q}(\mathbb{R}^N)$ for $q\in [1,N/|σ|)$. A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile.A by-product of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient.

math.AP↗
arXiv2024

Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles

Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $σ=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$ it is shown that, for $m+q>2$, there exists a unique exponent $β_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.

math.AP↗
arXiv2024

Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \ (0,\infty)\times\mathbb{R}^N$$ are studied when $N\geq1$ and the exponents $(p,q)$ satisfy $p_c=\frac{2N}{N+1}$, $p-1<q<\frac{p}{2}$. Existence and uniqueness of such a solution are established in dimension $N=1$. In dimension $N\geq2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to\infty$ is provided.

math.AP↗
arXiv2024

A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions

This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, $$ posed for $(x,t)\in\mathbf{R}^N\times(0,\infty)$, $N\geq1$, and in the range of exponents $1 0$. We give a complete classification of (singular) self-similar solutions of the form $$ u(x,t)=t^{-α}f(|x|t^{-β}), \ α=\frac{σ+2}{σ(m-1)+2(p-1)}, \ β=\frac{p-m}{σ(m-1)+2(p-1)}, $$ showing that their form and behavior strongly depends on the critical exponent $$ p_F(σ)=m+\frac{σ+2}{N}. $$ For $p\geq p_F(σ)$, we prove that all self-similar solutions have a tail as $ξ\to\infty$ of one of the forms $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)} \quad {\rm or} \quad u(x,t)\sim \left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-σ/(p-1)}, $$ while for $m<p<p_F(σ)$ we add to the previous the \emph{existence and uniqueness} of a \emph{compactly supported very singular solution}. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper.

math.AP↗
arXiv2024

Existence and multiplicity of blow-up profiles for a quasilinear diffusion equation with source

We classify radially symmetric self-similar profiles presenting finite time blow-up to the quasilinear diffusion equation with weighted source $$ u_t=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, $T>0$, in dimension $N\geq1$ and in the range of exponents $-2<σ<\infty$, $1<m<p<p_s(σ)$, where $$ p_s(σ)=\left\{\begin{array}{ll}\frac{m(N+2σ+2)}{N-2}, & N\geq3,\\ +\infty, & N\in\{1,2\},\end{array}\right. $$ is the renowned Sobolev critical exponent. The most interesting result is the \emph{multiplicity of two different types} of self-similar profiles for $p$ sufficiently close to $m$ and $σ$ sufficiently close to zero in dimension $N\geq2$, including \emph{dead-core profiles}. For $σ=0$, this answers in dimension $N\geq2$ a question still left open in \cite[Section IV.1.4, pp. 195-196]{S4}, where only multiplicity in dimension $N=1$ had been established. Besides this result, we also prove that, for any $σ\in(-2,0)$, $N\geq1$ and $m<p<p_s(σ)$ \emph{existence} of at least a self-similar blow-up profile is granted. In strong contrast with the previous results, given any $N\geq1$, $σ\geqσ^*=(mN+2)/(m-1)$ and $p\in(m,p_s(σ))$, \emph{non-existence} of any radially symmetric self-similar profile is proved.

math.AP↗
arXiv2024

Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=Δu^m + u^p, $$ posed in dimension $N\geq3$, $m>1$. More precisely, we show that there is always at least one solution in backward self-similar form if $p>p_s=m(N+2)/(N-2)$. In particular, this establishes \emph{non-optimality of the Lepin critical exponent} introduced in \cite{Le90} in the semilinear case $m=1$ and extended for $m>1$ in \cite{GV97, GV02}, for the existence of self-similar blow-up solutions. We also prove that there are multiple solutions in the same range, provided $N$ is sufficiently large. This is in strong contrast with the semilinear case, where the Lepin critical exponent has been proved to be optimal.

math.AP↗
arXiv2023

Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,T)$, with exponents $1 0$, are established. More precisely, we identify the \emph{optimal class of initial conditions} $u_0$ for which (local in time) existence is ensured and prove \emph{non-existence of solutions} for the complementary set of data. We establish then (local in time) \emph{uniqueness and a comparison principle} for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem \emph{blows up in a finite time} $T\in(0,\infty)$ and \emph{finite speed of propagation} holds true for $t\in(0,T)$: if $u_0\in L^{\infty}(\mathbb{R}^N)$ is an initial condition with compact support and blow-up time $T>0$, then $u(t)$ is compactly supported for $t\in(0,T)$. We also establish in this work the \emph{absence of localization at the blow-up time} $T$ for solutions stemming from compactly supported data.

math.AP↗
arXiv2023

Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles

Existence of specific \emph{eternal solutions} in exponential self-similar form to the following quasilinear diffusion 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 $σ=σ_c:=2(1-q)/(m-1)$ is proved. Looking for radially symmetric solutions of the form$$u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$we show that there exists a unique exponent $β^*\in(0,\infty)$ for which there exists a one-parameter family $(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $σ\in (0,σ_c)$.

math.AP↗
arXiv2023

Some qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=Δu^m+(1+|x|)^σu^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, where $m>1$, $p\in(0,1)$ and $σ>0$. Initial data are taken to be bounded, non-negative and compactly supported. In the range when $m+p\geq2$, we prove \emph{local existence of solutions} together with a \emph{finite speed of propagation} of their supports for compactly supported initial conditions. We also show in this case that, for a given compactly supported initial condition, there exist \emph{infinitely many solutions} to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range $m+p<2$, we establish new \emph{Aronson-Bénilan estimates} satisfied by solutions to the Cauchy problem, which are of independent interest as a priori bounds for the solutions. We apply these estimates to establish \emph{infinite speed of propagation} of the supports of solutions if $m+p<2$, that is, $u(x,t)>0$ for any $x\in\real^N$, $t>0$, even in the case when the initial condition $u_0$ was compactly supported.

math.AP↗
arXiv2023

Extinction and non-extinction profiles for the sub-critical fast diffusion equation with weighted source

We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, $N\geq3$, in the sub-critical range of the fast diffusion equation $0 0$ and $\max\{p_c(σ),1\} 0$, in the form $$ u(x,t)=t^αf(|x|t^β), \qquad f(ξ)\sim Cξ^{-(N-2)/m}, \qquad α>0, \ β>0 $$ exist, provided $0<m<m_s=(N-2)/(N+2)$ and $p_s(σ)=m(N+2σ+2)/(N-2)<p<p_L(σ)$. On the other hand, we prove that there exists $p_0(σ)\in(p_c(σ),p_s(σ))$ such that self-similar solutions presenting finite time extinction are established both for $p\in(p_0(σ),p_s(σ))$ and for $p\in(p_s(σ),p_L(σ))$, but with profiles $f(ξ)$ having different spatially decreasing tails as $|x|\to\infty$. We also prove non-existence of self-similar solutions in complementary ranges of exponents to the ones described above or if $m\geq m_c$.

math.AP↗
arXiv2022

Radial equivalence and applications to the qualitative theory for a class of non-homogeneous reaction-diffusion equations

Some transformations acting on radially symmetric solutions to the following class of non-homogeneous reaction-diffusion equations $$ |x|^{σ_1}\partial_tu=Δu^m+|x|^{σ_2}u^p, \qquad (x,t)\in\real^N\times(0,\infty), $$ which has been proposed in a number of previous mathematical works as well as in several physical models, are introduced. We consider here $m\geq1$, $p\geq1$, $N\geq1$ and $σ_1$, $σ_2$ real exponents. We apply these transformations in connection to previous results on the one hand to deduce general qualitative properties of radially symmetric solutions and on the other hand to construct self-similar solutions which are expected to be patterns for the dynamics of the equations, strongly improving the existing theory. We also introduce mappings between solutions which work in the semilinear case $m=1$.

math.AP↗