Search

This article explores blow-up phenomena in nonlinear logarithmic wave equations with fractional damping, infinite memory, and strong dissipation. The paper proves the existence of a local weak solution using semigroup theory. Furthermore, this research demonstrates that under certain conditions in finite time, the local solution may blow-up by using an appropriate Lyapunov functional. The findings highlight the effectiveness of strong damping, particularly when combined with fractional damping. [ABSTRACT FROM AUTHOR]

We consider the following chemotaxis-haptotaxis system: { u t = ∇ ⋅ (D (u) ∇ u) − χ ∇ ⋅ (S (u) ∇ v) − ξ ∇ ⋅ (u ∇ w) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 3 with smooth boundary. It is proved that for S (s) D (s) ≤ A (s + 1) α for α < 2 n and under suitable growth conditions on D , there exists a uniform-in-time bounded classical solution. Also, we prove that for radial domains, when the opposite inequality holds, the corresponding solutions blow-up in finite or infinite-time. We also provide the global-in-time existence and boundedness of solutions to the above system with small initial data when D (s) = 1 , S (s) = s. [ABSTRACT FROM AUTHOR]

In this paper, we mainly revisit a weakly dissipative periodic two‐component b$$ b $$‐family system. Considering the dissipative effect, the local well‐posedness is first obtained for the system by applying the Kato's semigroup theory. We then utilize the characteristics line method to get one blow‐up criterion with the dispersive parameter k1∈(1,3]$$ {k}_1\in \left(1,3\right] $$. Finally, the other blow‐up criterion is derived with regard as considering the odevity of initial data by using energy method. When the parameters λ,ki(i=1,2,3)$$ \lambda, {k}_i\left(i&#x0003D;1,2,3\right) $$ belong to suitable range, our obtained results supplement the corresponding blow‐up criteria, which was derived by Liu and Yin (Nonlinear Analysis: Real World Applications 12 (2011) 3608–3620). [ABSTRACT FROM AUTHOR]

In this paper, ihe influence of an additive white noise forcing term on ihe numerical solution for a class of deierminisiic nonlinear one-dimensional Schrödinger equaiions wiih mixed concave convex was studied, sub-super nonlinearities, that is, the stationary states and the blowing-up solutions. Such a perturbation occurs when the size of the noise, described by the real-value parameter e is positive. The size of the noise is controlled by the parameter e >0. We also proved that as e approaches zero, the solution of the perturbed problem converges to the unique trajectory of the deterministic equation, which is the solitary wave. The stochastic model appears to be more realistic, and one can observe, for small values of e, a similar evolution phenomena about the solution as that given by the deterministic case. However, an explosion of the solution and a blow-up phenomena can be noted as e becomes bigger. [ABSTRACT FROM AUTHOR]

In this paper, we consider a quasi-linear parabolic equation u t = u p (x xx + u) . It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p > 2 . We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2 < p < 3 and in the case of p = 3 , separately. In addition, we assert that if 2 ≤ p ≤ 3 then lim t ↗ T (T - t) 1 p + ε max u (x , t) = 0 z for any ε > 0 under some assumptions. [ABSTRACT FROM AUTHOR]

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation (− Δ) s u + V (x) u = u 2 s ∗ − 1 − 휖 in ℝ N , where 휖 > 0, s ∈ (0,1), 2 s ∗ : = 2 N N − 2 s and N > 4s, as we deal with finite energy solutions. We show that the ground state u휖 blows up and precisely with the following rate ∥ u 휖 ∥ L ∞ (ℝ N) ∼ 휖 − N − 2 s 4 s , as 휖 → 0 + . We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior. [ABSTRACT FROM AUTHOR]

Considered in this paper is a generalized Camassa-Holm equation, which includes both the Camassa-Holm equation and the Novikov equation as two special cases. Firstly, two blow-up criteria are established for the generalized Camassa-Holm equation. Then we derive two blow-up phenomena, where a new L2k estimate plays a crucial role. In addition, we also show that peakon solutions are global weak solutions.

In this paper, we investigate so-called slowly traveling wave solutions for a quasilinear parabolic equation in detail. Over the past three decades, the motion of the plane curve by the power of its curvature with positive exponent α has been intensively investigated. For this motion, blow-up phenomena of curvature on cusp singularity in the plane curve with self-crossing points have been studied by several authors. In their analysis, particularly in estimating the blow-up rate, the slowly traveling wave solutions played a significantly important role. In this paper, aiming to clarify the blow-up phenomena, we derive an asymptotic expansion of the slowly traveling wave solutions with respect to the parameter κ , which is proportional to the maximum of the curvature of the curve, as κ goes to infinity. We discovered that the result depends discontinuously on the parameter δ = 1 + 1 / α . It suggests that the blow-up phenomenon may also drastically change according to parameter δ . [ABSTRACT FROM AUTHOR]

The main intention of this paper is to deal with the weak singularity of solutions to some nonlinear Hadamard type fractional differential systems (HTFDSs) which could be viewed as the generalization of classic Hadamard fractional settings. Resorting to the lower and upper solutions technique and constructing the compatible weighted Banach space, the completely continuous operator described by Hadamard type fractional versions as well as a sufficient criterion for the existence of blow-up solutions to some nonlinear HTFDSs is established. Additionally, the comparative analysis on the blow-up rate affected by critical system parameters is also presented, and several examples clearly illustrate the effectiveness and efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to studying the blow-up phenomena and instability of the smooth solution for the isentropic Chaplygin gas equations in R N for any dimension N ≥ 1. We first give two blow-up phenomena of the Chaplygin gas equations, if the initial data satisfy some conditions (compact support or spherical symmetry). Next, the dynamics and instability of a family of solutions for the Cauchy problem of equation (1.5) is investigated, if the velocity u = c (t) r. [ABSTRACT FROM AUTHOR]

Abstract The two-component μ-Hunter–Saxton system is considered in the spatially periodic setting. Firstly, two wave-breaking criteria are derived by employing the transport equation theory and the localization analysis method. Secondly, a sufficient condition of the blow-up solutions is established by using the classic method. The results obtained in this paper are new and different from those in previous works.

The present article is concerned with the study of blow-up phenomena of the smoooth solutions for the full compressible Euler equations in R d , d ≥ 1 , which have always been a great concern to physicists and mathematicians throughout history. The approach is to construct exact explicit function to study singularities in finite time of the spherically symmetric solutions, provided the initial data satisfy some conditions. Compared with the results obtained by Sideris in 1984 (CMP), in this article, the initial velocity field is not required to have a compact support and the initial density and entropy is not equal to a constant outside the support of the initial velocity field. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system which includes the Fokas-Olver-Rosenau-Qiao (FORQ) or modified Camassa-Holm (MCH) equation and the two-component MCH system as two special cases. The system is integrable in the sense of possessing a non-isospectral Lax pair with the spectrum depending on time t , and admits multi-peakon solutions in an explicit form. Furthermore, we establish the local well-posedness for the system in the Besov space B 2 , r s (T) with s > 3 / 2 , 1 ≤ r ≤ ∞ , where the key ingredients include the Friedrichs regularization method, the Littlewood-Paley decomposition theory, and the transport theory in Besov spaces. Then we derive a precise blow-up criteria, which is dependent of the parameters α (t) and γ (t). Moreover, by the intrinsic structure of the system, we obtain a new blow-up result for strong solutions with sufficient conditions on the initial data and parameters. The entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves. [ABSTRACT FROM AUTHOR]

In this paper, we present a new wave-breaking criteria for the spatially periodic two-component Degasperis-Procesi system. Such criterion can be easily applied to check that the earlier blow-up criteria to this system (on the line case) also hold. [ABSTRACT FROM AUTHOR]

In the recent decades, the biological models have been noticed to find a suitable numerical procedure. Among the biological models, equations in tumor growth have many applications. In the current paper, we consider some equations in chemotaxis and haptotaxis models. The studied models have blow-up phenomena in their solutions. In addition, the proposed numerical technique is based on a meshless method that is well-known generalized smoothed particle hydrodynamics (SPH) method. [ABSTRACT FROM AUTHOR]

We consider generic 2 × 2 singular Liouville systems... {−Δu1 = 2λ1 eu1 − aλ2 eu2 − 2π(α1 − 2)δ0 in Ω, −Δu2 = 2λ2 eu2 − bλ1 eu1 − 2π(α2 − 2)δ0 in Ω, u1 = u2 = 0 on ∂Ω, where Ω Э 0 is a smooth bounded domain in R2 possibly having some symmetry with respect to the origin, δ0 is the Dirac mass at 0, λ1, λ2 are small positive parameters and a, b, α1, α2 > 0. We construct a family of solutions to (☆) which blow up at the origin as λ1 → 0 and λ2 → 0 and whose local mass at the origin is a given quantity depending on a, b, α1, α2. In particular, if ab < 4 we get finitely many possible blow-up values of the local mass, whereas if ab ≥ 4 we get infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

This paper studies the Cauchy problem for an integrable three-component Camassa–Holm system. We first establish the local well-posedness with initial condition in Besov spaces. Then we prove a blow-up criteria by arguing inductively with respect to the regularity index. Moreover, we derive a Riccati-type differential inequality by using the structure of equations, and also prove a new blow-up criteria with sufficient conditions on initial condition, whose proof is based on the conservative property of potential densities along the characteristic. [ABSTRACT FROM AUTHOR]

The initial‐boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating uid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in Bsp,r × Bs-1p,r with s > max{1+ 1/p, 3/2, 2-1/p}, p, r ∊ [1,∞] by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space B3/22,1 × B1/22,1, and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained. [ABSTRACT FROM AUTHOR]

Let ( M , g ) be an n -dimensional compact Riemannian manifold. Let h be a smooth function on M and assume that it has a critical point ξ ∈ M such that h ( ξ ) = 0 and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics g λ = u λ 4 n − 2 g , with u > 0 , whose scalar curvature is the prescribed function h λ : = λ 2 + h , where λ is a small parameter. In the positive case, i.e. when the scalar curvature R g is strictly positive, we find a family of “bubbling” metrics g λ , where u λ blows up at the point ξ and approaches zero far from ξ as λ goes to zero. In the general case, if in addition we assume that there exists a non-degenerate conformal metric g 0 = u 0 4 n − 2 g , with u 0 > 0 , whose scalar curvature is equal to h , then there exists a bounded family of conformal metrics g 0 , λ = u 0 , λ 4 n − 2 g , with u 0 , λ > 0 , which satisfies u 0 , λ → u 0 uniformly as λ → 0 . Here, we build a second family of “bubbling” metrics g λ , where u λ blows up at the point ξ and approaches u 0 far from ξ as λ goes to zero. In particular, this shows that this problem admits more than one solution. [ABSTRACT FROM AUTHOR]

In the paper, we mainly investigate blow-up phenomena and global existence of strong solutions to a two-component Camassa–Holm systems with an arbitrary smooth function H . For three types of smooth functions H , by using a conservation law and the sign-preserving property of strong solutions, we obtain two new blow-up results and a new global existence result for the two-component system. Our obtained results generalize and cover the recent results in Yan et al. (2015), Zhang and Yin (2015, 2016). [ABSTRACT FROM AUTHOR]

The initial-boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper considers the Cauchy problem and blow-up phenomena of a new integrable two-component Camassa–Holm system, which is a natural extension of the Fokas–Olver–Rosenau–Qiao equation. Firstly, the local well-posedness of the system in the critical Besov space B 2 , 1 1 2 ( R ) × B 2 , 1 1 2 ( R ) is investigated, and it is shown that the data-to-solution mapping is Hölder continuous. Then, a blow-up criteria for the Cauchy problem in the critical Besov space is derived. Moreover, with conditions on the initial data, a new blow-up criteria is obtained by virtue of the blow-up criteria at hand and the conservative property of m and n along the characteristics. Finally, a global existence result for the strong solution is established. [ABSTRACT FROM AUTHOR]

We consider the critical heat equation in where is a smooth bounded domain in and is a ball of of center and radius small. We show that if is small enough, then there exists a sign-changing stationary solution of (CH) such that the solution of (CH) with initial value blows up if is sufficiently small. This shows, in particular, that the set of the initial conditions for which the solution of (CH) is global and bounded is not star-shaped. [ABSTRACT FROM AUTHOR]

We mainly study the Cauchy problem of modified Camassa-Holm and Degasperis-Procesi equations. First, we establish the local well-posedness for the equation in Besov space. Secondly, we derive the conservation laws and a precise blow-up scenario. Moreover, we prove the existence of blow-up solutions and obtain its blow-up rate, provided the initial data satisfy certain conditions. Finally, we present the persistence properties of the equation. [ABSTRACT FROM AUTHOR]

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra–Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we discuss the relationships between necessary conditions for blow-up of collocation solutions and exact solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the finite time blow-up phenomena for the vector nonlinear Schrödinger equations with a magnetic field which describe the spontaneous generation of a magnetic field in a cold plasma in the subsonic limit. After obtaining some a priori estimates, we prove under certain natural conditions that the solutions to the Cauchy problem of the vector nonlinear Schrödinger equations in two and three spatial dimensions blow up in a finite time. Assuming that a solution to the aforementioned vector nonlinear Schrödinger equations is radially symmetric with respect to spatial variables x, we show that if the initial energy is non-positive, then the solution blows up in three dimensions in a finite time. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we prove that -norm of the vorticity controls the blow-up phenomena of strong solutions to the Navier–Stokes equations. Our criterion is a limiting case in Yuan-Zhang and an improvement of Giga’s result . [Copyright &y& Elsevier]

Abstract: We mainly study the Cauchy problem of the periodic generalized Degasperis–Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we mainly study the Cauchy problem of the generalized Degasperis–Procesi equation. We establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which blow up in finite time. [Copyright &y& Elsevier]

In this paper, we numerically investigate nonlinear Schro¨dinger equations with a stochastic contribution which is of white noise type and acts either as a potential (multiplicative noise) or as a forcing term (additive noise). In the subcritical case, we recover similar results as in the case of the Korteweg–de Vries equation. In the critical or supercritical case, we observe that depending on its smoothness, the noise may have different effects. Spatially smooth noises amplify blow-up phenomena, whereas delta correlated multiplicative noises prevent blow-up formation. Note that in this latter case, very few results are known, both from a theoretical and a numerical point of view. [Copyright &y& Elsevier]

We consider the weakly dissipative modified two-component Dullin–Gottwald–Holm (mDGH2) system. We derive a simple sufficient condition on initial date to guarantee the blow-up of the solutions in finite time to the weakly dissipative mDGH2 system. [ABSTRACT FROM AUTHOR]

The two-component μ-Hunter-Saxton system is considered in the spatially periodic setting. Firstly, two wave-breaking criteria are derived by employing the transport equation theory and the localization analysis method. Secondly, a sufficient condition of the blow-up solutions is established by using the classic method. The results obtained in this paper are new and different from those in previous works. [ABSTRACT FROM AUTHOR]