Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f(u)=u^p,0<p1, we gained blow-up rate estimate.     

On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

A Regularity Criterion for the Navier–Stokes Equations

We prove that a weak solution u = (u ¹, u ², u ³) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrin's criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L ^{6, ∞}.     

An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u₁ + au_x1 ? vδu = 0 in ?ⁿ × ?⁺ with initial data u⁰; the Support of u⁰ is contained in ?(x₁ < 0) and a: ?ⁿ → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?⁺, we propose the artificial boundary condition: u₁ + au_x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v²) for small values of the viscosity v.     

Classification of type I and type II behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|^p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y,s) converges to either φ^∗(y) or −φ^∗(y) as s→∞, where φ^∗ denotes the singular stationary solution; (c) u(x,T)/φ^∗(x) tends to ±1 as x→0, where T is the blow-up time.Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L¹ continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t−T)^1/(p−1)‖u(⋅,t)L^∞_‖ is bounded as t↘T. We also give various criteria for complete and incomplete blow-ups.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

On the Blow-up Criterion of Smooth Solutions to the 3D Ideal MHD Equations

Abstract In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if . This is an improvement of the result given by Caflisch, Klapper, and Steele [3].     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow

In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (?Δ)^αu in the space and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in [0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided . For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if , which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.     

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u_t=Δu+u^p+u^q(x^*,t) in B(R) where p,q>0,B(R)={x∈R^N:|x|<R} and x^*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.

(i)

If or p=q>2, then single point blow-up occurs whenever solutions blow up.

(ii)

If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.

(iii)

If p?1<q, total blow-up occurs whenever solutions blow up.

(iv)

If max(p,q)?1, every solution exists globally in time.

     

Uniqueness and stability of regional blow-up in a porous-medium equation

We study the blow-up phenomenon for the porous-medium equation in R^N, N1, u_t=Δu^m+u^m, m>1, for nonnegative, compactly supported initial data. A solution u(x,t) to this problem blows-up at a finite time . Our main result asserts that there is a finite number of points x₁,…,x_kR^N, with |x_i−x_j|2R^* for i≠j, such that Here w_*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation in R^N and R^* is the radius of its support. Moreover u(x,t) remains uniformly bounded up to its blow-up time on compact subsets of . The question becomes reduced to that of proving that the ω-limit set in the problem consists of a single point when its initial condition is nonnegative and compactly supported.     

On universality of blow-up profile for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> critical nonlinear Schrödinger equation

We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iu_t=-u-|u|^4/Nu with initial condition u₀H¹. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].We establish in this paper the existence of a universal blow-up profile which attracts blow-up solutions in the vicinity of blow-up time. Such a property relies on classification results of a new type for solutions to critical NLS. In particular, a new characterization of soliton solutions is given, and a refined study of dispersive effects of (NLS) in L² will remove the possibility of self similar blow-up in energy space H¹.     

Lower-semicontinuity result for the two-scale convergence

Let (m, n) ∈ ℕ², Ω an open bounded domain in ℝ^m , Y = [0, 1]^m ; u^ε in (L²(Ω))ⁿ which is two-scale converges to some u in (L²(Ω × Y))ⁿ . Let φ: Ω × ℝ^m × ℝⁿ → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝ^m × ℝⁿ , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C₁ > 0 and a function C₂ ∈ L²(Ω) such that

     

A regularity criterion for the solutions of 3D Navier-Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R³. More precisely, let u=(u¹,u²,u³) be a weak solution in (0,T)×R³. Then u becomes a classical solution if any two functions of _∂1u¹, _∂2u² and _∂3u³ belong to Lθ(0,T;Lr(R³)) provided with , .     

Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation

We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?² → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vⁿ, we obtain a sequence of spatially highly oscillatory classical solutions uⁿ. By considering the Young measures (YMs) ν and µ generated by the sequences vⁿ and uⁿ, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vⁿ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.     

Solution of a nonlinear heat equation with arbitrarily given blow-up points

We consider the equation where I ? ? u is scalar-valued and p > 1. It has been proven that if u(t) blows up at time T, the blow-up points are finite in number and located in I°. Our aim is to prove that this result is optimal. That is, for any given points x₁,…,x_k in I° there is a solution u such that its blow-up points are exactly x₁,…,x_k.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.