Search

In this paper, we consider the nonlinear cauchy problem 휗ρ + (−Δ)휃 2휗 = λȷ0/ρ1−αe휗 + μȷ0/ρ1−α|휗|p−1휗,x ∈ ℝN,ρ > 0,휗(x, 0) = 휗0(x), x ∈ ℝN, where J0/ραf(x) := − 1 Γ(α)∫0ρ f(s) (ρ−s)1−αds and 0 < 휃 ≤ 2, 0 < α < 1. We will study the equation for λ = μ = 1. We demonstrate the local existence and uniqueness of the solution to this problem by the Banach fixed-point principle. Then, it is shown that local solutions experience blow-up. Finally, we present the blow-up rate when 휃 = 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear p-k-Hessian systems with the gradient term where γ > 0, p > 1, and 1 ≤ k ≤ N is an integer. We show some existence, boundedness and blow-up results of the entire positive p-k-convex radial solutions based on the monotone iterative method. Finally, we present an example to illustrate our result. [ABSTRACT FROM AUTHOR]

This paper addresses the questions of well-posedness to fractional m-Laplacian reaction diffusion equation with singular potential term and logarithmic nonlinearity: x - 2 s ∂ t u + (- Δ) m s u + (- Δ) s ∂ t u = u | u | - 2 R (u) , where R (u) = u r ln (| u |) . Guided by the made assumptions, we arrive at the conclusions of the local and global solvability of solutions within the framework of Galerkin approximation. In addition, this study considers weak solutions' asymptotic stability and explosion in finite time. Significantly, we not only figure out the relationship between the non-local fractional operator and singular potential term, but generalize and improve earlier results in the literature. [ABSTRACT FROM AUTHOR]

For small-amplitude semilinear wave equations with power type nonlinearity on the first-order spatial derivative, the expected sharp upper bound on the lifespan of solutions is obtained for both critical cases and subcritical cases, for all spatial dimensions $ n>1 $. It is achieved uniformly by constructing the integral equations, deriving the ordinary differential inequality system, and iteration argument. Combined with the former works, the sharp lifespan estimates for this problem are completely established, at least for the spherical symmetric case. [ABSTRACT FROM AUTHOR]

The paper deals with a generalization of H. A. Levine's well-known energy method. Due to this generalization, we were able to consider various examples of nonlinear systems of equations and obtain sufficient conditions for the occurrence of blow-up. [ABSTRACT FROM AUTHOR]

In this work we consider a system of nonlinear Schrödinger equations whose nonlinearities satisfy a power-type-growth. First, we prove that the Cauchy problem is local and global well-posedness in L 2 and H 1 . Next, we establish the existence of ground state solutions. Then we use these solutions to study the dichotomy of global existence versus blow-up in finite time. Similar results were presented in the reference Noguera and Pastor (Commun Contemp Math 23:2050023, 2021. ) for the special case when the growth of the nonlinearities was quadratic. Here we will extend them to systems with nonlinearities of order p (cubic, quartic and so on). Finally, we recover some known results for two particular systems, one with quadratic and the other with cubic growth nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following fractional Kirchhoff‐type pseudo parabolic equation driven by a nonlocal integro‐differential operator ℒK$$ {\mathcal{L}}_K $$: ut|x|2s+M([u]s2)ℒKu+ℒKut=|u|p−2ulog|u|,$$ \frac{u_t}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{2s}}&#x0002B;M\left({\left[u\right]}_s&#x0005E;2\right){\mathcal{L}}_Ku&#x0002B;{\mathcal{L}}_K{u}_t&#x0003D;{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\log \mid u\mid, $$where [u]s$$ {\left[u\right]}_s $$ represents the Gagliardo seminorm of u$$ u $$. Instead of imposing specific assumptions on the Kirchhoff function, we introduce a more general sense to establish the local existence of weak solutions. Moreover, via the sharp fractional Hardy inequality, the decay estimates for global weak solutions, the blow‐up criterion, blow‐up rate, and the upper and lower bounds of the blow‐up time are derived. Lastly, we discuss the global existence and finite time blow‐up results with high initial energy. [ABSTRACT FROM AUTHOR]

We study a fourth‐order hyperbolic equation involving Kirchhoff type p$$ p $$‐Laplacian and superlinear source, subject to zero Navier boundary condition, utt+Δ2u−a+b‖∇u‖ppdiv(|∇u|p−2∇u)=c|u|q−2u,(x,t)∈Ω×(0,T),$$ {u}_{tt}&#x0002B;{\Delta}&#x0005E;2u-\left(a&#x0002B;b{\left\Vert \nabla u\right\Vert}_p&#x0005E;p\right)\operatorname{div}\left({\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;{p-2}\nabla u\right)&#x0003D;c{\left&#x0007C;u\right&#x0007C;}&#x0005E;{q-2}u,\left(x,t\right)\in \Omega \times \left(0,T\right), $$where Ω$$ \Omega $$ is an open bounded domain in ℝn$$ {\mathrm{\mathbb{R}}}&#x0005E;n $$ with n≥3$$ n\ge 3 $$; T(≤+∞)$$ T\left(\le &#x0002B;\infty \right) $$ denotes the maximal existence time; and a,b>0$$ a,b>0 $$ and p,q>2$$ p,q>2 $$ are constants. For q<2p$$ q<2p $$, using auxiliary function method and Sobolev inequality, we prove that there are only global solutions. For q≥2p$$ q\ge 2p $$, we obtain the optimal classification of initial energy and Nehari energy, which guarantees the existence of blow‐up solutions and global solutions. In the critical case q=2p$$ q&#x0003D;2p $$, we find out that the coefficients of the Kirchhoff term and the superlinear source play important role in separating out the property of weak solutions. [ABSTRACT FROM AUTHOR]

Periodic peaked waves of the Novikov equation have been proved to be H1$$ {H}&#x0005E;1 $$‐orbital stable. Utilizing the method of characteristics, we establish that the periodic peakons of the Novikov equation are linearly unstable under W1,∞(핋) perturbations. Moreover, it is proved that the small initial W1,∞(핋) perturbations of the above periodic peakons can lead to the blow‐up phenomenon in finite time in the nonlinear evolution of the Novikov equation. [ABSTRACT FROM AUTHOR]

We study a class of damped p$$ p $$‐ biharmonic$$ biharmonic $$ type wave equation with homogeneous Dirichlet boundary condition. First of all, we prove the local existence of weak solutions by Galerkin method. Besides, when the energy level is low E(0)

In this paper, we investigate the initial-boundary value problem for finitely degenerate pseudo-parabolic equation associated with the Hörmander type operator. We first specify the conditions under which the weak solutions of the equation exist globally and finite time blow up when the initial energy J (u 0) > d . Especially, the life span of the blow-up solution and the blow-up result with arbitrarily high initial energy level are derived. Additionally, we show the solution of degenerate pseudo-parabolic problem will converge to the steady state solution. Finally, we obtain the existence of ground state solution to the stationary problem. We prove that there exists initial value such that the instability occurs as a blow-up in finite time and the corresponding solution exists globally, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we delve into an initial-boundary value problem of the dispersive-dissipative wave equation having strong damping, $ u_{tt}-\Delta u-\Delta u_{t}+\Delta^{2} u-\Delta u_{tt} = a|u|^{p-1} u-b|u|^{q-1} u \quad \text{in}\ \Omega\times(0, T_{\max}), $where $ \Omega $ is an open bounded domain in $ \mathbb{R}^{n} $ with smooth boundary; $ T_{\max} $ is the maximal existence time; $ a, b>0 $, $ p, q>1 $ are constants. Through the combination of the contraction mapping principle and the Faedo-Galerkin method, we achieve the local existence and uniqueness of the weak solution. For $ p>q $ or $ p = q $ and $ a>b $, we obtain the existence of the global weak solution, and the energy decay estimate. Moreover, based on the auxiliary function approach, we demonstrate that the weak solutions blow up in finite time under distinct conditions related to the initial energy. Furthermore, we succeed in obtaining lower and upper bounds for blow-up time (life span). For $ p

This paper is devoted to the analysis of blow-up solutions for the generalized Hartree equations. To begin with, we derive the sharp threshold of global existence and blow-up. Next, we demonstrate the mass concentration properties and the limiting behavior of blow-up solutions. Furthermore, we establish the nonexistence of minimal mass blow-up solutions by using the limiting behavior of blow-up solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the hyperbolic Keller-Segel equations in H s (T d) on torus with d ≥ 1. Firstly, developing the dissipative mechanism through translation, we establish the global well-posedness in H s (T d) (s > 1 + d 2) with initial data near some equilibrium state. Secondly, by capturing the feature of the preservation of zero directional derivative, we give a class of initial date that lead to finite time blow-up. It's worth noting that our method of proving blow-up phenomenon does not require any conservation law. Finally, the characterization of this blow-up motivates us to show the ill-posedness of this system in H 3 2 (T d) in the sense of "norm inflation", which implies that our ill-posedness result for this system is sharp on one dimensional torus. [ABSTRACT FROM AUTHOR]

We consider the L 2 -supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. In that case, there are obvious non-scattering global solutions, i.e., the standing waves. In this paper, we show a scattering and blow-up dichotomy for threshold even solutions in the low-frequency case. We emphasize that this dichotomy still holds at the critical frequency between high and low. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problems for multi-term time-fractional wave equations in the framework of Besov spaces, which can be described the Couette flow of viscoelastic fluid. Considering the initial data in Besov spaces, we obtain some results about the local well-posedness and the blow-up of mild solutions for the proposed problem. Further, we extend these results to Besov–Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping ‖ut‖2qut$$ {\left\Vert {u}_t\right\Vert}_2&amp;#x0005E;q{u}_t $$ and nonlinear source term. The main purpose of this paper is to provide a systematic research on the dynamic behavior of the solutions with three different energy levels. More precisely, we prove the global existence, energy decay estimate and blow‐up of solution at both subcritical (E(0)

This work investigates the non‐global existence of solutions for the inhomogeneous nonlinear Schrödinger problem without gauge invariance. Indeed, initially, we establish the non‐global existence of certain mass‐subcritical solutions. Subsequently, we demonstrate the non‐global existence of energy‐subcritical solutions characterized by small initial data. These results are are acknowledged to be unsuccessful for the classical gauge invariant inhomogeneous nonlinear Schrödinger equation (INLS). [ABSTRACT FROM AUTHOR]

In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: γΔ2u−a+b∫ℝN|∇u|2dxΔu+u=|u|2σu,x∈ℝN,u∈H2(ℝN),$$ \left\{\begin{array}{c}\gamma {\Delta}&amp;#x0005E;2u-\left(a&amp;#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&amp;#x0005E;N}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;2\mathrm{d}x\right)\Delta u&amp;#x0002B;u&amp;#x0003D;{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{2\sigma }u,x\in {\mathrm{\mathbb{R}}}&amp;#x0005E;N,\\ {}u\in {H}&amp;#x0005E;2\left({\mathrm{\mathbb{R}}}&amp;#x0005E;N\right),\end{array}\right. $$where γ,a,b>0$$ \gamma, a,b&gt;0 $$ are constants and σ≥1$$ \sigma \ge 1 $$. We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for 1≤σ<4N−4$$ 1\le \sigma &lt;\frac{4}{N-4} $$ if N=5$$ N&amp;#x0003D;5 $$, 6, and 7, and for 1≤σ<+∞$$ 1\le \sigma &lt;&amp;#x0002B;\infty $$ if N≤4$$ N\le 4 $$. Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension N=3$$ N&amp;#x0003D;3 $$. [ABSTRACT FROM AUTHOR]

The initial boundary-value problem for a plate equation with nonlocal damping is considered. The local existence of solution is proved by the monotone operator theory with locally Lipschitz perturbation. By using the potential well theory, we get the global existence of solution. By Nakao's inequality, the decay estimate of polynomial type is obtained. We provide also the sufficient conditions of finite time blow-up of weak solutions with suitable conditions on the initial data by an ordinary differential inequality for an appropriately chosen functional. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the Cauchy problem for a weakly coupled system of two semi-linear structurally damped σ k -evolution equations, where σ k ≥ 1 for k = 1 , 2 . Our first purpose involves the proof of global (in time) existence of small data energy solutions by mixing additional L m k regularity for the data, where m k ∈ [ 1 , 2) . We want to point out that in some cases of choosing suitable parameters m k , with k = 1 , 2 , the obtained lower bound of one exponent p or q related to power nonlinearities on the right-hand sides is really smaller than the critical exponent, the so-called modified Fujita exponent. The second aim of this paper is to discuss a blow-up result for Sobolev solutions with any different fractional values of σ k ≥ 1 when m 1 = m 2 . [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature. [ABSTRACT FROM AUTHOR]

A nonlinear viscoelastic Kirchhoff-type equation with a source, Balakrishnan-Taylor damping, dispersion and delay terms is studied. We prove the blow-up of solutions under suitable hypotheses. [ABSTRACT FROM AUTHOR]

In this paper, a class of mixed pseudo-parabolic-Kirchhoff equation is considered. We focus on the interaction between the nonlocal diffusion and the nonstandard growth condition. More specifically, we establish a threshold result for the global solutions and finite time blow-up of the weak solution for low and critical initial energy. Meanwhile, we give the upper and lower bound estimates of blow-up time for the blow-up solution. And then we investigate the convergent relation between the global solutions and the ground-state solutions. Finally, we give some conditions to ensure the global existence or blow-up of solutions for the high initial energy. [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem for a coupled non-degenerate Kirchhoff-Heat system with nonlinear source term is considered. Due to the asymmetry of the structure of the problem studied, we have to develop more techniques to balance this asymmetry. Roughly speaking, we obtain the following conclusions.On the one hand, for low initial energy, with the help of the modified potential well method and differential inequality techniques, we use the multiplier method and Komornik inequality to prove that the energy of the solution decays exponentially to zero when the initial data starts from the stable set; In addition, when the initial data starts from the unstable set, we also find that the solution blows up in finite time, meanwhile we obtain an upper bound for the lifespan.On the other hand, for high initial energy, we first prove the $ L^{2} $ norm of the solution with perturbation term is monotonically increasing with respect to time variable. This helps us establish a substitute for unstable set. In the sequel, we prove that the solution blows up in finite time, an estimate of the upper bound for the lifespan of the solution is showed as well.Finally, to bypass this difficulty arising from the failure of the embedding inequality $ (H_0^1(\Omega) $ $ \hookrightarrow L^{2(p-1)}(\Omega)) $ for the supercritical case $ (p>\frac{2(n-1)}{n-2}) $ with space dimension $ n $, we apply an inverse Hölder inequality with correction constants derived from energy inequalities and the interpolation inequality and then obtain an explicit lower bound of the lifespan. [ABSTRACT FROM AUTHOR]

For the problem ∂ t k u - ∑ | α | = m ∂ α a α (x , t , u) ≥ f (| u |) in R + n + 1 = R n × (0 , ∞) , u (x , 0) = u 0 (x) , ∂ t u (x , 0) = u 1 (x) , … , ∂ t k - 1 u (x , 0) = u k - 1 (x) ≥ 0 , where u i ∈ L 1 , l o c (R n) and a α are Caratheodory functions such that | a α (x , t , ζ) | ≤ A | ζ | p , A , p = c o n s t > 0 , for almost all (x , t) ∈ R + n + 1 and for all ζ ∈ R , we obtain exact conditions on the function f guaranteeing that any global weak solution is identically zero. [ABSTRACT FROM AUTHOR]

We study the Euler-Bernoulli equations with time delay: utt+Δ2u−g1∗Δ2u+g2∗Δu+μ1ut(x,t)∣ut(x,t)∣m−2+μ2ut(x,t−τ)∣ut(x,t−τ)∣m−2=f(u),{u}_{tt}+{\Delta }^{2}u-{g}_{1}\ast {\Delta }^{2}u+{g}_{2}\ast \Delta u+{\mu }_{1}{u}_{t}\left(x,t){| {u}_{t}\left(x,t)| }^{m-2}+{\mu }_{2}{u}_{t}\left(x,t-\tau ){| {u}_{t}\left(x,t-\tau )| }^{m-2}=f\left(u), where τ\tau represents the time delay. We exhibit the blow-up behavior of solutions with both positive and nonpositive initial energy for the Euler-Bernoulli equations involving time delay.

In the present paper, we are concerned with the analysis of the asymptotic characteristics of solutions within a new class of nonlinear fractional p (x ,.) - Kirchhoff parabolic equations as follows: u t + M u s p (x , y) L s p (x , y) u = | u | σ (x) - 2 u log | u | , in U × (0 , T) , u (x , t) = 0 , in ∂ U × (0 , T) , u (x , 0) = u 0 (x) , in U. Using potential well theory, Gronwall’s inequality, and Komornik’s integral inequality, we give a classification of global existence and blow-up in finite time of solutions when the initial data satisfies different conditions. Moreover, we give two-sided estimates of asymptotic behavior when the diffusion term dominates the source. [ABSTRACT FROM AUTHOR]

In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.

This paper investigated the blow-up properties of solutions to the initial value problem for a fourth-order nonlinear parabolic equation with an exponential source term. By using an improved concavity method, we obtained upper and lower bound estimates for the blow-up time of the solution.

Compressible Euler equations with space-dependent damping in high dimensions Rn(n=2,3){{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system: −G1(x)+∇⋅G2→(x)=0,−G2→(x)+∇G1(x)=μG2→(x)(1+∣x∣)λ.\left\{\begin{array}{l}-{G}_{1}\left(x)+\nabla \cdot \overrightarrow{{G}_{2}}\left(x)=0,\\ -\overrightarrow{{G}_{2}}\left(x)+\nabla {G}_{1}\left(x)=\frac{\mu \overrightarrow{{G}_{2}}\left(x)}{{(1+| x| )}^{\lambda }}.\end{array}\right. This result generalizes the one in the literature from 1−D1-D to high dimension Rn(n=2,3){{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3).

A subset S of vertices of a graph G is a determining set for G if every automorphism of G is uniquely determined by its action on S. The determining number of a graph G is the smallest integer r such that G has a determining set of size r. In this paper, we study the determining number of edge-corona product, hierarchical product of graphs and the determining number of blow-up of some graphs. Also, we investigate the determining number of the zero divisor graph of the ring [Formula: see text], for some values of n.

This article is concerned with the qualitative properties for the Cauchy problem of a non-Newtonian filtration equation with a reaction source term and volumetric moisture content. On the basis of the slowly decaying behavior of volumetric moisture content, we establish new critical exponents that depend on the ratio of coefficients and exponent of volumetric moisture content. Meantime, under appropriate conditions, we show that the solution globally exists for small enough initial data and blows up in finite time for large enough or any nontrivial initial datum.

This work was concerned with the weakly coupled system of semi-linear wave equations with time dependent speeds of propagation, damping terms, and derivative nonlinear terms in generalized Einstein-de Sitter space-time on $ \mathbb{R}^n $. Under certain assumptions about the indexes $ k_1, \, k_2 $, coefficients $ \mu_1, \, \mu_2 $, and nonlinearity exponents $ p, \, q $, applying the iteration technique, finite time blow-up of local solutions to the small initial value problem of the coupled system was investigated. Blow-up region and upper bound lifespan estimate of solutions to the problem were established. Compared with blow-up results in the previous literature, the new ingredient relied on that the blow-up region of solutions obtained in this work varies due to the influence of coefficients $ k_1, \, k_2 $.

This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (Comm. Contemp. Math., 19 (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method.