Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation

We consider finite time blow up solutions to the critical nonlinear Schrödinger equation with initial condition u₀ H¹. Existence of such solutions is known, but the complete blow up dynamic is not understood so far. For initial data with negative energy, finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so-called log-log law has been proved in [10], [11]. We focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur. We establish the stability in energy space H¹ of the log-log upper bound exhibited in the negative energy case, and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions.     

Large Mass Global Solutions for a Class of L 1-Critical Nonlocal Aggregation Equations and Parabolic-Elliptic Patlak-Keller-Segel Models

We consider a class of L ¹ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the L ¹ scaling symmetry and we prove that all ‘sufficiently spread out’ initial data, even with supercritical mass, results in global, decaying solutions. In particular, we produce decaying solutions with arbitrary mass in cases for which finite time blow up solutions or non-decaying solutions are also known to exist for sufficiently large mass. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with supercritical mass blow up in finite time. The results with linear diffusion are proved using properties of the Fokker-Planck semi-group whereas the results with nonlinear diffusion are proved using a more interesting bootstrap argument coupling the entropy-entropy dissipation methods of the porous media equation together with higher L ^p estimates similar to those used in small-data and local theory for PKS-type equations.     

Blowup of solutions of the unsteady Prandtl's equation

We prove that for certain classes of compactly supported C^∞ initial data, smooth solutions of the unsteady Prandtl's equation blow up in finite time. © 1997 John Wiley & Sons, Inc.     

Numerical blow‐up for the porous medium equation with a source

We study numerical approximations of positive solutions of the porous medium equation with a nonlinear source, where m > 1, p > 0 and L > 0 are parameters. We describe in terms of p, m, and L when solutions of a semidiscretization in space exist globally in time and when they blow up in a finite time. We also find the blow‐up rates and the blow‐up sets, proving that there is no regional blow‐up for the numerical scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Global existence,blow up and asymptotic behaviour of solutions for nonlinear Klein–Gordon equation with dissipative term

We study the Cauchy problem of nonlinear Klein–Gordon equation with dissipative term. By introducing a family of potential wells, we derive the invariant sets and prove the global existence, finite time blow up as well as the asymptotic behaviour of solutions. In particular, we show a sharp condition for global existence and finite time blow up of solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

A global existence result for a semilinear scale‐invariant wave equation in even dimension

In the present article a, semilinear scale‐invariant wave equation with damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension n is proved by using weighted L^∞ ? L^∞ estimates, under the assumption that the multiplicative constants, which appear in the coefficients of damping and of mass terms, fulfill an interplay condition, which yields somehow a “wave‐like” model. In particular, combining this existence result with a recently proved blow‐up result, a suitable shift of Strauss exponent is proved to be the critical exponent for the considered model. Moreover, the still open part of a conjecture done by D'Abbicco‐Lucente‐Reissig is proved to be true in the massless case.     

Blow‐up phenomena in chemotaxis systems with a source term

This paper deals with a parabolic–parabolic Keller–Segel‐type system in a bounded domain of , {N = 2;3}, under different boundary conditions, with time‐dependent coefficients and a positive source term. The solutions may blow up in finite time t^?; and under appropriate assumptions on data, explicit lower bounds for blow‐up time are obtained when blow up occurs. Copyright © 2015 John Wiley & Sons, Ltd.     

The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases

This paper concerns the threshold of global existence and finite time blow up of solutions to the time-dependent focusing Gross-Pitaevskii equation describing the Bose-Einstein condensation of trapped dipolar quantum gases. Via a construction of new cross-constrained invariant sets, it is shown that either the corresponding solution globally exists or blows up in finite time according to some appropriate assumptions about the initial datum.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations u_t=Δu+e^αtv^p and v_t=Δv+e^βtu^q, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley & Sons, Ltd.     

Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation

A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theory (1) is studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T^? < ∞ , with locally bounded final time profiles u(x, T^?) , are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T , all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) (2) created by Oleinik in the middle of the 1950s. Contrary to (1) and not surprisingly, self‐similar gradient blow‐up for (2) is shown to admit a unique continuation. Bearing in mind the classic form (2) , the NDE (1) can be written as (3) with the standard linear integral operator (?D²_x)^?1 > 0 . However, because (3) is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of (3) is increasing for BV data u₀(x) ).     

Global existence and nonexistence of solutions for the nonlinear pseudo‐parabolic equation with a memory term

In this paper, we study the initial boundary value problem of nonlinear pseudo‐parabolic equation with a memory term with initial conditions and Dirichlet boundary conditions. By the combination of the Galerkin method and Potential well theory, the existence of global solutions is derived. Moreover, not only the finite time blow up of solutions with the negative initial energy (E(0) < 0) but also the finite time blow up results with the nonnegative initial energy (0≤E(0) < d_k) are obtained by using Concavity method and Potential well theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical Study of Blow‐Up Mechanisms for Davey–Stewartson II Systems

We present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and that the time‐dependent scaling behaves as in the Ozawa solution and not as in the stable blow‐up of standard L ² critical nonlinear Schrödinger equation. The blow‐up profile is given by a dynamically rescaled lump.     

Explicit solutions of the viscous model vorticity equation

Explicit solutions are found for the viscous version of the model vorticity equation recently proposed by P. Constantin, P. D. Lax, and A. Majda: where H(w) is the Hilbert transform of w, and v is a positive constant. Various properties of these solutions, including the fact that they blow up after a finite time, are discussed.     

On the blow‐up rate for the heat equation with a nonlinear boundary condition

In this paper positive solutions of the heat equation with a nonlinear Neumann boundary conditions in an upper halfspace are studied. The optimal result on blow‐up rate, valid for all solutions which blow up in finite time, is established under the assumption that the exponent of a nonlinear boundary condition is subcritical in the Sobolev sense. This complements results derived for the bounded domain case in [10, 13] either for monotonous solutions or under a stronger restriction on the exponent of a boundary condition. Copyright © 2000 John Wiley & Sons, Ltd.     

Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities

In this article we investigate the possibility of finite time blow-up in H¹(R²) for solutions to critical and supercritical nonlinear Schrödinger equations with an oscillating nonlinearity. We prove that despite the oscillations some solutions blow up in finite time. Conversely, we observe that for a given initial data oscillations can extend the local existence time of the corresponding solution.     

Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R² in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.     

Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition

Semilinear hyperbolic and parabolic initial–boundary value problems are studied. Criteria for solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and general boundary condition to blow up in finite time are obtained.     

On the Well-Posedness for Stochastic Schr¨odinger Equations with Quadratic Potential

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schrödinger equations. The local and global well-posedness are proved with values in the space Σ(? ⁿ ) = {f ∈ H ₁(? ⁿ ), |·|f ∈ L ²(? ⁿ )}. When the nonlinearity is focusing and L ²-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If ? = 0, our results coincide with the ones for the deterministic equation.