Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBlow-up and global existence for the inhomogeneous porous medium equation with reaction
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term. We show that for small enough initial data, if $rho(x)sim frac{1}{left(log|x|right)^{alpha}|x|^{2}}$ as $|x|to infty$, then solutions globally exist for any $p>1$. On the other hand, when $rho(x)simfrac{left(log|x|right)^{alpha}}{|x|^{2}}$ as $|x|to infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $p>m$. Such results generalize those established in [27] and [28], where it is supposed that $rho(x)sim |x|^{-q}$ for $q>0$ as $|x|to infty$.
rate research
Read More
We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $rho(x) u^p$ with $p>1$; this is a mathematical model of a thermal evolution of a heated plasma (see [25]). The density decays slowly at infinity, in the sense that $rho(x)lesssim |x|^{-q}$ as $|x|to +infty$ with $qin [0, 2).$ We show that for large enough initial data, solutions blow-up in finite time for any $p>1$. On the other hand, if the initial datum is small enough and $p>bar p$, for a suitable $bar p$ depending on $rho, m, N$, then global solutions exist. In addition, if $p<underline p$, for a suitable $underline pleq bar p$ depending on $rho, m, N$, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypotehsis that $qin [0, epsilon)$ for $epsilon>0$ small enough, when $mleq p<underline p$. Observe that $underline p=bar p$, if $rho(x)$ is a multiple of $|x|^{-q}$ for $|x|$ large enough. Such results are in agreement with those established in [41], where $rho(x)equiv 1$. The case of fast decaying density at infinity, i.e. $qgeq 2$, is examined in [31].
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {it fast} at infinity, in the sense that $rho(x)sim |x|^{-q}$ as $|x|to +infty$ with $qge 2.$ In the case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when $q>2$, we show that existence of global in time solutions always prevails. The case of {it slowly} decaying density at infinity, i.e. $qin [0,2)$, is examined in [41].
The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly non-unique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones.
In this paper we consider the inhomogeneous nonlinear Schrodinger equation $ipartial_t u +Delta u=K(x)|u|^alpha u,, u(0)=u_0in H^s({mathbb R}^N),, s=0,,1,$ $Ngeq 1,$ $|K(x)|+|x|^s| abla^sK(x)|lesssim |x|^{-b},$ $0<b<min(2,N-2s),$ $0<alpha<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $alpha_0(b)<alpha<(4-2b)/N$. The value $alpha_0(b)$ is the positive root of $Nalpha^2+(N-2+2b)alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=mu|x|^{-b}$, $muin mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.
The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter $a$ which controls the strength of advection. For solutions on the infinite domain we find a new critical value $a_c=0.6890665337007457ldots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457ldots here then ldots doesnt fit into the line that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We find a new exact analytical collapsing solution at $a=1/2$ as well as prove the existence of a leading order complex singularity for general values of $a$ in the analytical continuation of the solution from the real spatial coordinate into the complex plane. This singularity controls the leading order behaviour of the collapsing solution. For $a_c<aleq1$, we find a blow-up solution in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For $a gtrsim 1.3$, we find that the solution exists globally with exponential-like growth of the solution amplitude in time. We also consider the case of periodic boundary conditions. We identify collapsing solutions for $a<a_c$ which are similar to the real line case. For $a_c<ale0.95$, we find new blow-up solutions which are neither expanding nor collapsing. For $ age 1,$ we identify a global existence of solutions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
