相关论文: Global solutions versus finite time blow-up for th…
Global self-similar solutions to the parabolic Hardy-H\'enon equation $$ u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $\sigma>\max\{-2,-N\}$. The…
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 $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed in dimension…
We classify the self-similar solutions presenting finite time blow-up to the parabolic Hardy-H\'enon equation $$ \partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and the range of…
Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $N\geq1$ and exponents $$…
We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, $N\geq3$,…
A new transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,\infty)$ and with…
We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=\Delta u^m + u^p, $$ posed in dimension $N\geq3$, $m>1$. More precisely, we show that there is always…
We classify radially symmetric self-similar profiles presenting finite time blow-up to the quasilinear diffusion equation with weighted source $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, $T>0$, in dimension…
We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for $x\in\real$,…
We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for…
We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=\Delta…
We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1\leq p<m$ and…
Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-H\'enon equation involving a quasilinear degenerate diffusion and a space-dependent superlinear source featuring a singular potential…
We prove existence and uniqueness of self-similar solutions with exponential form $$ u(x,t)=e^{\alpha t}f(|x|e^{-\beta t}), \qquad \alpha, \ \beta>0 $$ to the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta…
We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$ u_t-\Delta u =|v|^{p}+t^\sigma w_1(x),\,\, v_t-\Delta v…
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 $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, $$ posed for…
We consider the semilinear heat equation $$ u_t-\Delta u=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling…
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=\Delta u^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in…
The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$,…
We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of…