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Register a new userBlow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density
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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].
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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 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 $|x|to infty$.
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.
Let $OmegasubsetR^n$ be a smooth bounded domain and let $a_1,a_2,dots,a_{i_0}inOmega$, $widehat{Omega}=Omegasetminus{a_1,a_2,dots,a_{i_0}}$ and $widehat{R^n}=R^nsetminus{a_1,a_2,dots,a_{i_0}}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=Delta u^m$, $u>0$, in $widehat{Omega}times (0,infty)$ ($widehat{R^n}times (0,infty)$ respectively) which satisfies $u(x,t)toinfty$ as $xto a_i$ for any $t>0$ and $i=1,cdots,i_0$, when $0<m<frac{n-2}{n}$, $ngeq 3$, and the initial value satisfies $0le u_0in L^p_{loc}(2{Omega}setminus{a_1,cdots,a_{i_0}})$ ($u_0in L^p_{loc}(widehat{R^n})$ respectively) for some constant $p>frac{n(1-m)}{2}$ and $u_0(x)ge lambda_i|x-a_i|^{-gamma_i}$ for $xapprox a_i$ and some constants $gamma_i>frac{2}{1-m},lambda_i>0$, for all $i=1,2,dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,dots,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively) for some constant $mu_0>0$ and $gamma_1>frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $widehat{Omega}$ (or $widehat{R^n}$ respectively) to infinity as $ttoinfty$. If $u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively) for some constant $mu_0>0$ and satisfies $lambda_i|x-a_i|^{-gamma_i}le u_0(x)le lambda_i|x-a_i|^{-gamma_i}$ for $xapprox a_i$ and some constants $frac{2}{1-m}<gamma_ilegamma_i<frac{n-2}{m}$, $lambda_i>0$, $lambda_i>0$, $i=1,2,dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $2{Omega}setminus{a_1,a_2,dots,a_{i_0}}$ (or $widehat{R^n}$ respectively) to a harmonic function as $ttoinfty$.
We prove that any sufficiently differentiable space-like hypersurface of ${mathbb R}^{1+N} $ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation $partial_{tt} u - Delta u=|u|^{p-1} u$ on ${mathbb R} times {mathbb R} ^N$, for any $1leq Nleq 4$ and $1 < p le frac {N+2} {N-2}$. We follow the strategy developed in our previous work [arXiv 1812.03949] on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blowup on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at $t=0$ for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at $t=0$. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with $H^2times H^1$ solutions for the transformed problem.
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