We study the existence and uniqueness of solutions of $\partial_tu-Δu+u^q=0$ ($q>1$) in $Ω\times (0,\infty)$ where $Ω\subset\mathbb R^N$ is a domain with a compact boundary, subject to the conditions $u=f\geq 0$ on $\partialΩ\times (0,\infty)$ and the initial condition $\lim_{t\to 0}u(x,t)=\infty$. By means of Brezis' theory of maximal monotone operators in Hilbert spaces, we construct a minimal solution when $f=0$, whatever is the regularity of the boundary of the domain. When $\partialΩ$ satisfies the parabolic Wiener criterion and $f$ is continuous, we construct a maximal solution and prove that it is the unique solution which blows-up at $t=0$.
We present a blow-up rate estimate for a solution to the parabolic Gross-Pitaevskii and related systems on entire space with Sobolev subcritical nonlinearity. We extend the results of [Y. Giga, S. Matsui and S. Sasayama, Indiana Univ. Math. J. {53} (2004), 483--514] to the parabolic systems.
In this paper, we establish blow-up rates for higher-order semilinear parabolic equations with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.
It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.
We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.
In this paper we consider initial boundary value problem for nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition and nonnegative initial datum. We prove comparison principle, global existence and blow-up of solutions.
In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in $\mathbb{T}\times\mathbb{R}$ with advection by a large strictly monotone shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8π$: solutions with mass less than $8π$ are global in time and there exist solutions with mass larger than $8 π$ which blow up in finite time \cite{Schweyer14}. We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In contrast with the parabolic-elli…
In this paper, we consider the solution of a nonlocal parabolic equation. Focusing on the solutions with initial data at high energy level, we find the criteria for global existence and finite time blow up for the corresponding solution respectively. Moreover, we prove that there always exists blow up solution with negative Nehari functional no matter how large the energy is.
Let $G=(V,E)$ be a locally finite connected weighted graph, $Δ$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=Δu + f(u)$ on $G$. The blow-up phenomenons of the equation are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if $f$ satisfies appropriate conditions, then the solution of the equation blows up in a finite time.
In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the solutions to blow up in finite time. In particular, we obtain the existence of finite time blow-up solutions with arbitrary high initial energy. We also derive the upper bound and lower bound of the blow up time.
The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.
In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. Firstly, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions satisfy $J(u_0)\leq d$ and $I(u_0)\neq 0$, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increase exponentially.
We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data which are, in a suitable sense, smaller than the diffusion parameter $τ$ in the equation for the chemoattractant, we obtain global solutions, and for some data larger than $τ$ , a finite time blowup. In this way, we check that our size condition for the global existence is sharp for large $τ$ , up to a logarithmic factor.
We study positive blowing-up solutions of systems of the form: $$u_t=δ_1 Δu+e^{pv},\quad v_t= δ_2Δv+e^{qu},$$ with $δ_1,δ_2>0$ and $p, q>0$. We prove single-point blow-up for large classes of radially decreasing solutions. This answers a question left open in a paper of Friedman and Giga~(1987), where the result was obtained only for the equidiffusive case $δ_1=δ_2$ and the proof depended crucially on this assumption.
This paper concerns the initial-boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then prove the existence, uniqueness and decay estimate of global solutions and the blowup phenomena of solutions with subcritical initial energy. Next, we extend parallelly these results to the critical initial energy. Lastly, the existence, uniqueness and asymptotic behavior of global solutions with supercritical initial energy are proved by further analyzing …
In this paper, we consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.
This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing results, we delete the assumption that the solutions to stochastic heat equations are non-negative. Then the blowup phenomenon in the whole space is obtained by using the properties of heat kernel. We obtain that the solutions will blow up in finite time for nontrivial initial data.
We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v_1) + ξ\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = Δv_1 - βv_1 + αu \\ \partial_t v_2 = Δv_2 - δv_2 + γu, \end{cases} \end{equation*} posed with homogeneous Neumann boundary conditions in bounded domains $Ω=B_R \subset \mathbb{R}^3$, $R>0$, admits radially symmetric solutions which blow-up in finite time if it is attraction-dominated in the sense that $χα-ξγ>0$.
The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hop…
We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to …
This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=Δu-χ\nabla \cdot (u\nabla v)+ξ\nabla\cdot(u \nabla w), \quad v_t=Δv-v+u, \quad w_t=Δw-w+u, \quad x \in Ω,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $χ, ξ>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the …
We consider a parabolic-elliptic chemotaxis system generalizing \[ \begin{cases}\begin{split} & u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u(u+1)^{σ-1}\nabla v)\\ & 0 = Δv - v + u \end{split}\end{cases} \] in bounded smooth domains $Ω\subset \mathbb{R}^N$, $N\ge 3$, and with homogeneous Neumann boundary conditions. We show that *) solutions are global and bounded if $σ<m-\frac{N-2}N$ *) solutions are global if $σ\le 0$ *) close to given radially symmetric functions there are many initial data producing unbounded solutions if $σ>m-\frac{N-2}N$. In particular, if $σ\le 0$ and $σ> m-\frac{…
We consider a Parabolic-Elliptic system of PDE's with a chemotactic term in a $N$-dimensional unit ball describing the behavior of the density of a biological species "$u$" and a chemical stimulus "$v$". The system includes a nonlinear chemotactic coefficient depending of ``$\nabla v$", i.e. the chemotactic term is given in the form $$- div (χu |\nabla v|^{p-2} \nabla v), \qquad \mbox{ for } \ p \in ( \frac{N}{N-1},2), \qquad N >2 $$ for a positive constant $χ$ when $v$ satisfies the poisson equation $$- Δv = u - \frac{1}{|Ω|} \int_Ω u_0dx.$$ We study the radially symmetric solutions under the…
We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u Δu + u \int_Ω|\nabla u|^2 \] in bounded domains $Ω\subset\mathbb{R}^n$ and prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. We show that in this case the blow-up set coincides with $\overlineΩ$, i.e. the finite-time blow-up is global. Key words: Degenerate diffusion, non-local nonlinearity, blow-up, evolutionary games, infinite dimensional replicator dynamics
We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-Δu\leq v^λ\] \[0\leq v_t-Δv\leq u^σ\] in $Ω\times (0,1)$, where $λ$ and $σ$ are nonnegative constants and $Ω$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $λ$ and $σ$ such that solutions of this system satisfy pointwise bounds in compact subsets of $Ω$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.
