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Register a new userLower semicontinuous obstacles for the porous medium equation
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We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.
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We study a kind of generalized porous medium equation with fractional Laplacian and abstract pressure term. For a large class of equations corresponding to the form: $u_t+ u Lambda^{beta}u= ablacdot(u abla Pu)$, we get their local well-posedness in Fourier-Besov spaces for large initial data. If the initial data is small, then the solution becomes global. Furthermore, we prove a blowup criterion for the solutions.
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 singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d $le$ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.
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.
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].
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