Fast Diffusion Equation Research Articles

Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction

We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term posed in with N ⩾ 3, where and the critical value for the weight The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a change of sign of both self-similar exponents at m = m s = (N − 2)/(N + 2), leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a perfect equilibrium in the competition between the fast diffusion and the reaction effects.

Singular extinction profiles of solutions to some fast diffusion equations

We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.

Extinction behavior for the fast diffusion equations with critical exponent and Dirichlet boundary conditions

For a smooth bounded domain Ω ⊆ R n $\Omega \subseteq \mathbb {R}^n$ , n ⩾ 3 $n\geqslant 3$ , we consider the fast diffusion equation with critical sobolev exponent ∂ w ∂ τ = Δ w n − 2 n + 2 \begin{equation*} \hspace*{7pc}\frac{\partial w}{\partial \tau } =\Delta w^{\frac{n-2}{n+2}}\hspace*{-7pc} \end{equation*} under Dirichlet boundary condition w ( · , τ ) = 0 $w(\cdot , \tau ) = 0$ on ∂ Ω $\partial \Omega$ . Using the parabolic gluing method, we prove existence of an initial data w 0 $w_0$ such that the corresponding solution has extinction rate of the form ∥ w ( · , τ ) ∥ L ∞ ( Ω ) = γ 0 ( T − τ ) n + 2 4 ln ( T − τ ) n + 2 2 ( n − 2 ) ( 1 + o ( 1 ) ) \begin{equation*} \hspace*{13pc}\Vert w(\cdot , \tau )\Vert _{L^\infty (\Omega )} = \gamma _0(T-\tau )^{\frac{n+2}{4}}{\left|\ln (T-\tau )\right|}^{\frac{n+2}{2(n-2)}}(1+o(1)) \end{equation*} as t → T − $t\rightarrow T^-$ , here T > 0 $T > 0$ is the finite extinction time of w ( x , τ ) $w(x, \tau )$ . This generalizes a result of Galaktionov and King [30] for the radially symmetric case Ω = B 1 ( 0 ) : = { x ∈ R n | | x | < 1 } ⊂ R n $\Omega =B_1(0) : = \lbrace x\in \mathbb {R}^n||x| < 1\rbrace \subset \mathbb {R}^n$ .

Solutions with snaking singularities for the fast diffusion equation

We construct solutions of the fast diffusion equation, which exist for all t ∈ R t\in \mathbb {R} and are singular on the set Γ ( t ) ≔ { ξ ( s ) ; − ∞ &gt; s ≤ c t } \Gamma (t)≔\{ \xi (s) ; -\infty &gt;s \leq ct \} , c &gt; 0 c&gt;0 , where ξ ∈ C 3 ( R ; R n ) \xi \in C^3(\mathbb {R};\mathbb {R}^n) , n ≥ 2 n\geq 2 . We also give a precise description of the behavior of the solutions near Γ ( t ) \Gamma (t) .

Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations

We investigate the well-posedness of the fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness of solutions for L 1 L^1 initial data was established in Bonforte, Grillo, and Vázquez [J. Evol. Equ. 8 (2008), pp. 99–128]. However, in the Euclidean space, it is known from Herrero and Pierre [Trans. Amer. Math. Soc. 291 (1985), pp. 145–158] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in L l o c 1 L^1_{\mathrm {loc}} . We establish here that such data still give rise to global solutions on general manifolds. If, moreover, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to − ∞ -\infty at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, assuming in addition that the initial datum is in L l o c 2 L^2_{\mathrm {loc}} and nonnegative, a minimal solution is shown to exist, and we establish uniqueness of purely (nonnegative) distributional solutions, a fact that to our knowledge was not known before even in the Euclidean space. The required curvature bound is sharp, since on model manifolds it is equivalent to stochastic completeness, and it was shown in Grillo, Ishige, and Muratori [J. Math. Pures Appl. (9) 139 (2020), pp. 63–82] that uniqueness for the FDE fails even in the class of bounded solutions when stochastic completeness does not hold. A crucial ingredient of the uniqueness result is the proof of nonexistence of nonnegative, nontrivial distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest.

Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted Lp-path spaces is proved. In particular, as special cases the classical Caputo derivative and other fractional derivatives appearing in applications are included. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of typeddt(k⁎u)(t)+A(t,u(t))=f(t),0<t<T, with (in general nonlinear) operators A(t,⋅) satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on [0,∞) with lims→∞k(s)=0. Analogous results for the case, where f is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators and the time-fractional (stochastic) p-Laplace equation are covered.

Global Existence and Extinction Singularity for a Fast Diffusive Polytropic Filtration Equation with Variable Coefficient

In this article, we deal with an inhomogeneous fast diffusive polytropic filtration equation. By using the energy estimate approach, Hardy–Littlewood–Sobolev inequality, and a series of ordinary differential inequalities, we prove the global existence result and obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.

Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise

We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu and Röckner (2011) [4]], [Barbu and Röckner (2018) [6]], and [Gess (2014) [11]].

Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space

&lt;p style='text-indent:20px;'&gt;For &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n\ge 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 0&amp;lt;m&amp;lt;\frac{n-2}{n} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \beta&amp;lt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \alpha = \frac{2\beta}{1-m} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of the form &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; where &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ f_{\lambda} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a radially symmetric function satisfying&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;with &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, for some constant &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \lambda&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with initial value &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ u_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; satisfying &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ \forall x\in\mathbb{R}^n\setminus\{0\} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, such that the solution &lt;inline-formula&gt;&lt;tex-math id="M16"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; satisfies &lt;inline-formula&gt;&lt;tex-math id="M17"&gt;\begin{document}$ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M18"&gt;\begin{document}$ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, for some constants &lt;inline-formula&gt;&lt;tex-math id="M19"&gt;\begin{document}$ \lambda_1&amp;gt;\lambda_2&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We also prove the asymptotic large time behaviour of such singular solution &lt;inline-formula&gt;&lt;tex-math id="M20"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; when &lt;inline-formula&gt;&lt;tex-math id="M21"&gt;\begin{document}$ n = 3,4 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M22"&gt;\begin{document}$ \frac{n-2}{n+2}\le m&amp;lt;\frac{n-2}{n} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; holds. Asymptotic large time behaviour of such singular solution &lt;inline-formula&gt;&lt;tex-math id="M23"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is also obtained when &lt;inline-formula&gt;&lt;tex-math id="M24"&gt;\begin{document}$ 3\le n&amp;lt;8 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M25"&gt;\begin{document}$ 1-\sqrt{2/n}\le m&amp;lt;\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, and &lt;inline-formula&gt;&lt;tex-math id="M26"&gt;\begin{document}$ u(x,t) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is radially symmetric in &lt;inline-formula&gt;&lt;tex-math id="M27"&gt;\begin{document}$ x\in\mathbb{R}^n\setminus\{0\} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for any &lt;inline-formula&gt;&lt;tex-math id="M28"&gt;\begin{document}$ t&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; under appropriate conditions on the initial value &lt;inline-formula&gt;&lt;tex-math id="M29"&gt;\begin{document}$ u_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

Critical curves for a fast diffusive p-Laplacian equation with nonlocal source

ABSTRACT This paper deals with the critical curves for a fast diffusive p-Laplacian equation with nonlocal inner source under nonlinear boundary flux in half-line. We obtain new critical global existence curve and critical Fujita curve by constructing various self-similar super- and sub-solutions.

Higher integrability in the obstacle problem for the fast diffusion equation

We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case m<1. The result holds for the natural range of exponents that is known from other regularity results for porous medium type equations. We also cover the case of signed solutions.

Large deviation principle of occupation measures for non-linear monotone SPDEs

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises.

Fast diffusion in a porous building material with a nonlocal source

Abstract Fast diffusion is of great importance in concrete solidification, and this paper suggests a modified Fick law to deduce a fast diffusion equation with a nonlocal source. Its global and non-global solutions are obtained, and the critical Fujita exponent and second critical exponent are discussed. A numerical example is given to elucidate the basic properties of the fast diffusion process. The theoretical results shed a promising light on an extremely fast solidification of a concrete in building an artificial island or an urgent repairing in an accident.

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

We investigate the homogeneous Dirichlet problem for the fast diffusion equation ut = Δum, posed in a smooth bounded domain Ω ⊂ ℝN, in the exponent range ms = (N − 2)+/(N + 2) &lt; m &lt; 1. It is known that bounded positive solutions extinguish in a finite time T &gt; 0, and also that they approach a separate variable solution u(t, x) ∼ (T − t)1/(1 − m)S(x) as t → T−, where S belongs to the set of solutions to a suitable elliptic problem and depends on the initial datum u0. It has been shown recently that v(x, t) = u(t, x) (T − t)−1/(1 − m) tends to S(x) as t → T−, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behavior and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on an (improved) weighted Poincaré inequality, which we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of “almost‐orthogonality,” which can be thought of as a nonlinear analogue of the classical orthogonality condition needed to obtain improved Poincaré inequalities and sharp convergence rates for linear flows. © 2019 Wiley Periodicals, Inc.

Hydrodynamic limit for a facilitated exclusion process

We study the hydrodynamic limit for a periodic $1$-dimensional exclusion process with a dynamical constraint, which prevents a particle at site $x$ from jumping to site $x\pm1$ unless site $x\mp1$ is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than $\frac{1}{2}$, or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density $\frac{1}{2}$, the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

Super fast vanishing solutions of the fast diffusion equation

We will extend a recent result of B. Choi, P. Daskalopoulos and J. King [ 5 ]. For any \begin{document}$ n\ge 3 $\end{document} , \begin{document}$ 0 and \begin{document}$ \gamma>0 $\end{document} , we will construct subsolutions and supersolutions of the fast diffusion equation \begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document} in \begin{document}$ \mathbb{R}^n\times (t_0, T) $\end{document} , \begin{document}$ t_0 , which decay at the rate \begin{document}$ (T-t)^{\frac{1+\gamma}{1-m}} $\end{document} as \begin{document}$ t\nearrow T $\end{document} . As a consequence we obtain the existence of unique solution of the Cauchy problem \begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document} in \begin{document}$ \mathbb{R}^n\times (t_0, T) $\end{document} , \begin{document}$ u(x, t_0) = u_0(x) $\end{document} in \begin{document}$ \mathbb{R}^n $\end{document} , which decay at the rate \begin{document}$ (T-t)^{\frac{1+\gamma}{1-m}} $\end{document} as \begin{document}$ t\nearrow T $\end{document} when \begin{document}$ u_0 $\end{document} satisfies appropriate decay condition.

The critical exponent for fast diffusion equation with nonlocal source

This paper considers the Cauchy problem for fast diffusion equation with nonlocal source u_{t}=Delta u^{m}+ (int_{mathbb{R}^{n}}u^{q}(x,t),dx )^{frac{p-1}{q}}u^{r+1}, which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent p_{c}=m+frac{2q-n(1-m)-nqr}{n(q-1)}, namely, any solution of the problem blows up in finite time whenever 1< ple p_{c}, and there are both global and non-global solutions if p>p_{c}.

Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

This work is concerned with the gradient flow of absolutely $p$-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite ($p<2$) or infinite extinction time ($p \geq 2$). We give upper bounds for the finite extinction time and establish convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and nonlocal versions of PDEs like $p$-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting. We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.

Critical exponents for the fast diffusion equation with a nonlinear boundary condition

In this paper we consider the fast diffusion equation ∂tu=Δ(um) (x∈Ω, t>0) with a nonlinear boundary condition ∂νum=up (x∈∂Ω, t>0), where 0<m<1, p>0, Ω⊂RN is a smooth domain and N≥1. We prove that p0=(m+1)/2 is the critical global existence exponent for the cases Ω=RN∖B1‾ (N≥2) and Ω=B1:={x∈RN:|x|<1} (N≥1).

New critical exponents, large time behavior, and life span for a fast diffusive p-Laplacian equation with nonlocal source

This paper mainly investigate positive solutions of the Cauchy problem for a fast diffusive p-Laplacian equation with nonlocal source $$\begin{aligned} u_{t}=\Delta _pu+\left( \,\,\int \limits _{{\mathbb {R}}^N}u^q(y,t)\mathrm{d}y\right) ^{\frac{r-1}{q}}u^{s+1},\quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{aligned}$$ where $$N\ge 1$$ , $$\frac{2N}{N+1}<p<2$$ , $$q>1$$ , $$r\ge 1$$ , $$0\le s<\left( 1+\frac{1}{N}\right) p-2$$ and $$r+s>1$$ . We obtain the new critical Fujita exponent by virtue of the auxiliary function method and forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.