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Diana-Rodica Munteanu

Publications and source records attributed to Diana-Rodica Munteanu.

4 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

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 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↗