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Register a new userOn the generalized porous medium equation in Fourier-Besov spaces
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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.
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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.
In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r}, s>1+frac{1}{p}, 1leq p,rleq+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$, which improves the previous work cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r}, s>1+frac{1}{p}, 1leq pleq+infty, 1leq r<+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$. At last, we show that the solution is ill-posed in $B^{sigma}_{p,infty}$ with $sigma>3+frac{1}{p}, 1leq pleq+infty$.
We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ partial_t u + (-Delta)^{ 1/2} u = | abla u|^p, quad x in mathbb R^N, t > 0, qquad u(x,0) = u_0(x) , quad x in mathbb R^N, $$ where $p > 1$ and $u_0 in B^1_{r,1}(mathbb R^N) cap B^1_{infty,1} (mathbb R^N)$ with $r in [1,infty]$. We show that for sufficiently small $u_0 in dot B^1_{infty,1}(mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.
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.
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