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Razvan Gabriel Iagar

Publications and source records attributed to Razvan Gabriel Iagar.

At least 19 recordsLinked to original sources
arXiv2026

A porous medium equation with dominating weighted absorption: three types of self-similar solutions

Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ with exponents $1 0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-α}f(|x|t^β), \quad α=\frac{σ+2}{σ(m-1)+2(p-1)}, \quad β=\frac{m-p}{σ(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \lim\limits_{ξ\to\infty}ξ^{σ/(p-1)}f(ξ)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a \emph{dead-core}; that is, $f\equiv0$ for $ξ\in[0,ξ_0]$ for some $ξ_0>0$, and, finally, there exists $K^*\in(0,\infty)$ such that, for any $K\in(0,K^*)$, there is at least a solution such that $$ \lim\limits_{ξ\to0}ξ^{-(σ+2)/(m-p)}f(ξ)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.

math.AP↗
arXiv2026

New sharp inequalities involving non-relative, relative and cross informational functionals with some remarkable minimizers of generalized Gaussian and Beta types

Several new and sharp informational inequalities are derived as a byproduct of Stam-like and moment-entropy-like inequalities in the relative framework and a recently established inequality mixing the Rényi entropy, the Rényi divergence and the Rényi cross entropy of suitable probability density functions. More precisely, we obtain a Stam-like inequality connecting the Rényi entropy power, the recently introduced scaling-invariant relative Fisher information and the Rényi cross entropy. Furthermore, we derive an inequality involving only Fisher-like informational measures and another inequality involving only moment-like functionals of non-relative, relative and cross types, respectively. All the inequalities are sharp. The minimizers of the Stam-like inequality are, in certain cases, pairs of Gaussian or stretched Gaussian probability densities; in contrast, each minimizer of the moment-like inequality is the probability density of the generalized Beta distribution.

cs.IT↗
arXiv2026

Convergence to self-similarity for a degenerate parabolic equation with fast-growing spatially-dependent absorption

The large time behavior of non-negative solutions to the absorption-diffusion equation $\partial$\textsubscr{t} u = $Δ$ u\textsuperscript{m} - |x|\textsuperscript{$σ$ }u\textsuperscript{m} in (0,$\infty$) x \textsuperscript{N}, with m > 1 and $σ$> $σ$\textsubscr{0}\,:= N (m-1)/(m+1) is identified. It is shown that all solutions approach a unique stationary solution in self-similar variables, which also provides a universal upper bound (friendly giant ), strongly contrasting to the standard case $σ$ = 0. On the one hand, the convergence proof exploits the variational structure of the equation and a suitable Caffarelli-Kohn-Nirenberg inequality, along with the B{é}nilan-Crandall homogeneity regularizing effect. On the other hand, the detailed study of the stationary problem combines elliptic estimates, Moser iteration and techniques from ordinary differential equations.

math.AP↗
arXiv2026

Large time behavior and transition from vanishing to spreading regimes for the generalized Burgers-Fisher-KPP equation

The large time behavior of solutions to the following generalized Burgers-Fisher-KPP equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k\in\mathbb{R}$, is considered in this work. Denoting by $H(x,t)$, respectively $\widetilde{H}(x,t)$ the solutions having as initial condition the Heaviside, respectively the ``anti-Heaviside" functions $$ H_0(x)=\begin{cases} 0, & \mbox{if } x<0 1, & \mbox{if } x\geq0. \end{cases}, \quad \widetilde{H}_0(x)=1-H_0(x), $$ critical velocities $\overline{c}$, respectively $\widetilde{c}=kn+2\sqrt{p-q}$, are identified such that $H(x,t)$, respectively $\widetilde{H}(x,t)$ approach the unique traveling wave solution of the equation with these critical velocities as $t\to\infty$. The critical velocity $\overline{c}$ is \emph{anomalous}, that is, it cannot be made explicit by an algebraic expression. Assuming for simplicity $k>0$, a remarkable fact is that, while $\widetilde{H}(x,t)\to0$ as $t\to\infty$ uniformly on compact subsets of $\mathbb{R}$, the Heaviside solution $H$ might tend either to zero or to one as $t\to\infty$, depending on the sign of the critical velocity $\overline{c}$. This sign vary with respect to the exponents $n$, $p$, $q$ and the coefficient $k$ and, in fact, we prove that given $p$, $q$, $n$, there exists a critical coefficient $k^*(n,p,q)$ such that $\overline{c}>0$ if $k>k^*(n,p,q)$ and $\overline{c}<0$ if $k<k^*(n,p,q)$. The convergence to either zero or one reflects the sharp influence of the convection term, since in the absence of it (that is, $k=0$), $H(x,t)$ would always tend to zero as $t\to\infty$. The results include more general initial conditions than the Heaviside-type functions, and sharp estimates of the threshold coefficient $k^*(n,p,q)$ are also given.

math.AP↗
arXiv2026

Blow-up rates and sets for a quasilinear diffusion equation with weighted source

Blow-up rates are established for general solutions to the quasilinear diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ in the range of exponents $1 0$. More precisely, if we consider a compactly supported solution $u(x,t)$ with blow-up time $T=T(u)\in(0,\infty)$, we derive the blow-up rate $$ C_1(T-t)^{-α}\leq \|u(x,t)\|_{\infty}\leq C_2(T-t)^{-α}, \quad t\in(0,T), $$ for some positive constants $C_1$, $C_2$, and the upper rate of expansion of the support $$ \sup\{|x|:u(x,t)>0\}\leq C_0(T-t)^{-β}, \quad t\in(0,T), $$ for some constant $C_0>0$, where $$ α=\frac{σ+2}{L}, \quad β=\frac{m-p}{L}, \quad L=σ(m-1)+2(p-1). $$ We also analyze the blow-up sets of solutions $u$, showing, under a suitable condition, that either $B(u)=\mathbb{R}^N$ or blow-up takes place only as $|x|\to\infty$.

math.AP↗
arXiv2026

Entropies, cross-entropies and Rényi divergence: sharp three-term inequalities for probability density functions

A new sharp inequality featuring the differential Rényi entropy, the Rényi divergence and the Rényi cross-entropy of a pair of probability density functions is established. The equality is reached when one of the probability density function is an escort density of the other. This inequality is applied, together with a general framework of a pair of transformations reciprocal to each other, to derive a number of further inequalities involving both classical and new informational functionals. A remarkable fact is that, in all these inequalities, the Rényi divergence of two probability density functions is sharply bounded by quotients of informational functionals of cross-type and single type. More precisely, we derive sharp inequalities composed by relative and cross versions of the absolute moments, or of the Fisher information measures (among others), and involving two and three probability density functions.

cs.IT↗
arXiv2026

Global self-similar solutions for Hardy-Hénon equations with linear and quasilinear diffusion

Global self-similar solutions to the parabolic Hardy-Hénon equation $$ u_t=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $σ>\max\{-2,-N\}$. The classification varies strongly with respect to the celebrated \emph{Fujita} and \emph{Sobolev critical exponents} $$ p_F(σ)=m+\frac{σ+2}{N}, \quad p_S(σ)= \begin{cases} \frac{m(N+2σ+2)}{N-2}, & \mbox{if } N\geq3, \\[1mm] \infty, & \mbox{if } N\in\{1,2\}. \end{cases} $$ Indeed, if $p\in(p_F(σ),p_S(σ))$, both equations admit self-similar solutions with either compact support (if $m>1$) or Gaussian-like tail as $|x|\to\infty$ (if $m=1$), as well as a one-parameter family satisfying $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)}, \quad {\rm as} \ |x|\to\infty. $$ If $p\geq p_S(σ)$, there are only self-similar solutions with the latter algebraic tail, while for $m<p\leq p_F(σ)$ no global solutions exist. The results open the way for a deeper study of the role of these solutions in the dynamics of the Hardy-Hénon equations.

math.AP↗
arXiv2026

Self-similar extinction for a fast diffusion equation with weighted absorption

Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $N\geq1$ and exponents $$ p>1, \quad m_c=\frac{(N-2)_+}{N} σ_*:=\frac{2(p-1)}{1-m}, $$ is established. Moreover, the existence of self-similar solutions of the form $$ U(x,t)=(T-t)^αf(|x|(T-t)^β), \quad α=\frac{σ+2}{(1-m)(σ-σ_*)}, \ β=\frac{p-m}{(1-m)(σ-σ_*)}, $$ with $f(0)>0$, $f'(0)=0$ and $$ \lim\limits_{ξ\to\infty}ξ^{(σ+2)/(p-m)}f(ξ)=L\in(0,\infty). $$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $σ=0$), where the strict positivity of solutions for any $t\in(0,\infty)$ is well-known.

math.AP↗
arXiv2025

A new group of transformations related to the Kullback-Leibler and Rényi divergences and universal classes of monotone measures of statistical complexity

In this work we introduce a family of transformations, named \textit{divergence transformations}, interpolating between any pair of probability density functions sharing the same support. We prove the remarkable property that the whole family of Kullback-Leibler and Rényi divergences evolves in a monotone way with respect to the transformation parameter. Moreover, fixing the reference density, we show that the divergence transformations enjoy a group structure and can be derived through the algebraic conjugation of the recently introduced differential-escort transformations and their relative counterparts. This algebraic structure allows us to deform any density function in such a way its divergence with respect a fixed reference density might also increase as much as possible. We also establish the monotonicity of composed measures involving the proper Kullback-Leibler and Rényi divergences as well as other recently introduced relative measures of moment and Fisher types. As applications, an approximation scheme of general density functions by simple functions is provided. In addition, we give a number of analytical and numerical examples of interest in both regimes of increasing and decreasing divergence.

math-ph↗
arXiv2025

Generalized informational functionals and new monotone measures of statistical complexity

In this paper we introduce a biparametric family of transformations which can be seen as an extension of the so-called up and down transformations. This new class of transformations allows to us to introduce new informational functionals, which we have called \textit{down-moments} and \textit{cumulative upper-moments}. A remarkable fact is that the down-moments provide, in some cases, an interpolation between the $p$-th moments and the power Rényi entropies of a probability density. We establish new and sharp inequalities relating these new functionals to the classical informational measures such as moments, Rényi and Shannon entropies and Fisher information measures. We also give the optimal bounds as well as the minimizing densities, which are in some cases expressed in terms of the generalized trigonometric functions. We furthermore define new classes of measures of statistical complexity obtained as quotients of the new functionals, and establish monotonicity properties for them through an algebraic conjugation of up and down transformations. All of these properties highlight an intricate structure of functional inequalities.

cs.IT↗
arXiv2025

Self-similar blow-up solutions for the supercritical parabolic Hardy-Hénon equation

We classify the self-similar solutions presenting finite time blow-up to the parabolic Hardy-Hénon equation $$ \partial_tu=Δu+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and the range of exponents $$ σ\in(-2,\infty), \quad p>p_S(σ):=\frac{N+2σ+2}{N-2}. $$ We establish the \emph{existence of self-similar blow-up solutions for any $p>p_S(σ)$}, provided $σ\geq2$. Moreover, we prove that, if $k$ is any natural number and $σ\geq 4k-2$, the parabolic Hardy-Hénon equation has at least $k$ different self-similar blow-up solutions for any $p>p_S(σ)$. These results are in a stark contrast with the standard reaction-diffusion equation $$ \partial_tu=Δu+u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ for which non-existence of any self-similar solution has been established, provided $p$ overpasses the Lepin exponent $p_L:=1+\frac{6}{N-10}$, $N\geq11$. For $σ\in(-2,2)$, we derive the expression of generalized Lepin exponents $p_L(σ)$ for $σ\in(0,2)$, respectively $\overline{p_L}(σ)$ for $σ\in(-2,0)$, and prove existence of self-similar solutions with finite time blow-up for $p\in(p_S(σ),p_L(σ))$, respectively $p\in(p_S(σ),\overline{p_L}(σ))$. Numerical evidence of the optimality of these exponents is also included.

math.AP↗
arXiv2025

Traveling wave solutions for the generalized Burgers-Fisher equation

Traveling wave solutions, in the form $u(x,t)=f(x+ct)$, to the generalized Burgers-Fisher equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k>0$, are classified with respect to their speed $c\in(-\infty,\infty)$ and the behavior at $\pm\infty$. The existence and uniqueness of traveling waves with any speed $c\in\mathbb{R}$ is established and their behavior as $x\to\pm\infty$ is described. In particular, it is shown that there exists a unique $c^*\in(0,\infty)$ such that there exists a unique soliton $f^*$ with speed $c^*$ and such that $$ \lim\limits_{ξ\to-\infty}f^*(ξ)=\lim\limits_{ξ\to\infty}f^*(ξ)=0, \quad ξ=x+ct. $$ Moreover, if $n p+q+1$ then $c^*>kn$. For $c<\min\{c^*,kn\}$, any traveling wave with speed $c$ satisfies $\lim\limits_{ξ\to-\infty}f(ξ)=0$ and $\lim\limits_{ξ\to\infty}f(ξ)=1$, while for $c>\max\{c^*,kn\}$ any traveling wave with speed $c$ satisfies $\lim\limits_{ξ\to-\infty}f(ξ)=1$ and $\lim\limits_{ξ\to\infty}f(ξ)=0$. In particular, for any speed $c\in(0,c^*)$, there are traveling wave solutions $u$ with speed $c$ such that $u(x,t)\to1$ as $t\to\infty$, in contrast to the non-convective case $k=0$.

math.AP↗
arXiv2025

A porous medium equation with spatially inhomogeneous absorption. Part II: Large time behavior

We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $σ>0$ and with initial conditions either satisfying $$ u_0\in L^{\infty}(\real^N)\cap C(\real^N), \quad \lim\limits_{|x|\to\infty}|x|^θu_0(x)=A\in(0,\infty) $$ for some $θ\geq0$. A number of different asymptotic profiles are identified, and uniform convergence on time-expanding sets towards them is established, according to the position of both $p$ and $θ$ with respect to the following critical exponents $$ p_F(σ)=m+\frac{σ+2}{N}, \quad θ_*=\frac{σ+2}{p-m}, \quad θ^*=N. $$ More precisely, solutions in radially symmetric self-similar form decaying as $|x|\to\infty$ with the rates $$ u(x,t)\sim A|x|^{-θ_*}, \quad {\rm or} \quad u(x,t)\sim \left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-σ/(p-1)}, $$ are obtained as asymptotic profiles in some of these cases, while asymptotic simplifications or logarithmic corrections in the time scales also appear in other cases. The uniqueness of some of these self-similar solutions, left aside in the first part of this work, is also established.

math.AP↗
arXiv2025

Sharp non-existence threshold for a parabolic Hardy-H{é}non equation with quasilinear diffusion

Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{é}non equation $$ \partial\_tu=Δu^m+|x|^σu^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, $$ with $m>0$, $σ>0$ and $p>\max\{1,m\}$, are identified. Assuming that the initial condition satisfies $$ u\_0\in L^{\infty}(\mathbb{R}^N), \quad \lim\limits\_{|x|\to\infty}|x|^γu\_0(x)=L\in(0,\infty), \quad u\_0\geq0, $$ it is shown that non-existence of solution occurs for $$ γ<\frac{σ+2}{p-m} - \frac{2\max{\{p-p\_G,0\}}}{(p-1)(p-m)} $$ with $$ p\_G:=1+\frac{σ(1-m)}{2}. $$ The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of $γ$.

math.AP↗
arXiv2025

Sharp informational inequalities involving Kullback-Leibler and Rényi divergences and a family of scaling-invariant relative Fisher measures

We introduce a new transformation called \emph{relative differential-escort}, which extends the usual differential-escort transformation by relating the change of variable to a reference probability density. As an application of it, we define a biparametric family of \emph{relative Fisher measures} presenting significant advantages with respect to the pre-existing ones in the literature: invariance under scaling changes and, consequently, sharp inequalities between the new relative Fisher measure and the well established Kullback-Leibler and Rényi divergences. We also introduce a biparametric family of \emph{relative cumulative moment-like measures} and we establish sharp lower bounds of these new measures by the Kullback-Leibler and Rényi divergences. The optimal bound and the minimizing densities are given. We also construct a family of inequalities for an arbitrary and fixed minimizing density in which the so-called generalized trigonometric functions plays a central role, providing thus one more interesting application of the newly introduced inequalities and measures.

math-ph↗
arXiv2025

Through and beyond moments, entropies and Fisher information measures: new informational functionals and inequalities

We introduce new classes of informational functionals, called \emph{upper moments}, respectively \emph{down-Fisher measures}, obtained by applying classical functionals such as $p$-moments and the Fisher information to the recently introduced up or down transformed probability density functions. We extend some of the the most important informational inequalities to our new functionals and establish optimal constants and minimizers for them. In particular, we highlight that, under certain constraints, the generalized Beta probability density maximizes (or minimizes) the upper-moments when the moment is fixed. Moreover, we apply these structured inequalities to systematically establish new and sharp upper bounds for the main classical informational products such as moment-entropy, Stam, or Cramér-Rao like products under certain regularity conditions. Other relevant properties, such as regularity under scaling changes or monotonicity with respect to the parameter, are studied. Applications to related problems to the Hausdorff moment problem are also given.

math-ph↗
arXiv2025

Large time behavior for a quasilinear diffusion equation with weighted source

The large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term $$ \partial_tu=Δu^m+\varrho(x)u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $m>1$, $1<p<m$ and suitable functions $\varrho(x)$, is established. More precisely, we consider functions $\varrho\in C(\mathbb{R}^N)$ such that $$ \lim\limits_{|x|\to\infty}(1+|x|)^{-σ}\varrho(x)=A\in(0,\infty), $$ with $σ\in(\max\{-N,-2\},0)$ such that $L:=σ(m-1)+2(p-1)<0$. We show that, for all these choices of $\varrho$, solutions with initial conditions $u_0\in C(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)\cap L^r(\mathbb{R}^N)$ for some $r\in[1,\infty)$ are global in time and, if $u_0$ is compactly supported, present the asymptotic behavior $$ \lim\limits_{t\to\infty}t^{-α}\|u(t)-V_*(t)\|_{\infty}=0, $$ where $V_*$ is a suitably rescaled version of the unique compactly supported self-similar solution to the equation with the singular weight $\varrho(x)=|x|^σ$: $$ U_*(x,t)=t^αf_*(|x|t^{-β}), \qquad α=-\frac{σ+2}{L}, \quad β=-\frac{m-p}{L}. $$ This behavior is an interesting example of \emph{asymptotic simplification} for the equation with a regular weight $\varrho(x)$ towards the singular one as $t\to\infty$.

math.AP↗