Alessandro Palmieri, Hiroyuki Takamura
In this paper, we consider a weakly coupled system of a wave and damped Klein-Gordon equation with nonlinearities of derivative type. We prove a blow-up result for the Cauchy problem associated with this system for nonnegative and compactly supported data by means of an iteration argument.
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In this note we try to understand the blow-up of solutions to Nakao's problem by using nonlinear ordinary differential inequalities.
In this paper, we consider blow-up behavior of weak solutions to a weakly coupled system for a semilinear damped wave equation and a semilinear wave equation in $\mathbb{R}^n$. This problem is part of the so-called Nakao's problem proposed by Professor Mitsuhiro Nakao (Kyushu University) for a critical relation between the exponents $p$ and $q$. By applying an iteration method for unbounded multipliers with a slicing procedure, we prove blow-up of weak solutions for Nakao's problem even for small data. We improve the blow-up result and upper bound estimates for lifespan comparing with the prev…
In the present paper, we investigate blow-up and lifespan estimates for a class of semilinear hyperbolic coupled system in $\mathbb{R}^n$ with $n\geqslant 1$, which is part of the so-called Nakao's type problem weakly coupled a semilinear damped wave equation with a semilinear wave equation with nonlinearities of derivative type. By constructing two time-dependent functionals and employing an iteration method for unbounded multiplier with slicing procedure, the results of blow-up and upper bound estimates for the lifespan of energy solutions are derived. The model seems to be hyperbolic-like i…
Blow-up results for a Nakao-type problem with a time-dependent damping term and derivative-type nonlinearities
该研究针对含导数型幂非线性耦合的半线性阻尼波动方程组，在尺度不变与产生散射两种时变阻尼情形下，利用双泛函迭代论证证明解的爆破并给出寿命上界。
On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term
本文研究含时变阻尼的半线性弱耦合波动方程组，利用迭代论证得到爆破结果与寿命上界，并揭示尺度不变及散射情形下爆破区域的差异。
In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schrödinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type. Especially, we find a sequence of initial data with mass approximating that of the ground state from above, the correspondng solution of which blows up. This result partially gives a positive answer to the open problem left by T.Tao, M. Visan and X. Zhang in \cite{TVZ}. After then, we obtain the lower blow-up rate and concentration rate for the blow-up solutions in the s…
Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities
本文针对两类含二次时间导数非线性项的一维非线性波动模型，构造分类五参数广义自相似爆破解族，证明相关分支渐近稳定性，统一刻画其爆破机制。
This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities \[ i\partial_t u-(-Δ)^su+λ_1|u|^{2p_1}u+λ_2|u|^{2p_2}u=0, \] where $0<p_1<p_2<\frac{2s}{N-2s}$. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $0<p_1<\frac{2s}{N}$ and $p_2=\frac{2s}{N}$. Finally, we investigate the dynam…
Nondegeneracy and asymptotic behavior of type I blow-up for semilinear heat equations with general nonlinearities
该研究针对具有一般非线性项的半线性热方程I型爆破解，通过拟相似变换建立局部一致收敛性，在弱假设下证明内部爆破点归一化解的正下界，覆盖多种非线性项情形。
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.
In this work, we show the existence of ground state solutions for an $l$-component system of non-linear Schrödinger equations with quadratic-type growth interactions in the energy-critical case. They are obtained analyzing a critical Sobolev-type inequality and using the concentration-compactness method. As an application, we prove the existence of blow-up solutions of the system without the mass-resonance condition in dimension six (and five), when the initial data is radial.
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convecti…
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and Glassey conjecture. In some cases, we obtain existence results, where the lower bound of the lifespan agrees with the upper bound in order. In addition, our results apply for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign co…
We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}Δu=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where $m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition b…
Blow-up criteria for the 1D NLS with combined nonlinearities
本文提出了一种用于一维NLS中结合非线性有限时间爆炸解的负能量和正能量准则，并给出了满足这些准则的初始数据示例。
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities \[ iu_{t}+Δu=λ_1|u|^{p_1}u+λ_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and $0<p_2<\frac{4}{N}$, we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for this equation. This is a complement to the result of Tao et al. (Comm. Partial Differential Equations 32: 1281-1343, 2007). Moreover, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rates and limiting p…
In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find …
Blow-up and sharp lifespan estimates to the weakly coupled system of structurally damped wave equations with critical nonlinearities
研究弱耦合结构阻尼波方程组中非线性项模连续性的临界曲线，通过分析小数据Sobolev解的全局存在性和有限时间爆破现象，确定模连续性的精确条件及局部解的寿命估计。
We calculate the full asymptotic expansion of boundary blow-up so- lutions, for any nonlinearity f . Our approach enables us to state sharp qualitative results regarding uniqueness and ra- dial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. At last, we study in more detail the standard nonlinearities f (u) = u^p, p > 1
Blow-up solutions to a class of nonlinear coupled Schrödinger systems with power-type-growth nonlinearities
该研究将二次增长非线性项耦合薛定谔方程组的爆破相关结果推广至p阶幂型增长情形，证明了适定性、基态解存在性，分析了全局存在与有限爆破二分性，并还原了两类特例的已知结论。
Finite time blow-up and global solutions for the viscoelastic wave equation with combined power-type nonlinearities
本文研究了具有组合幂型非线性项的粘弹性波方程的有限时间爆炸和全局解问题，分析了能量耗散与源项的平衡对解的存在性和衰减率的影响。
In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=Δu-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0, &u(x,0)=u_0(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\geq 0$ and $W$ is even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$ is a complex-valued function of $x$. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of…
We calculate the full asymptotic expansion of boundary blow-up solutions, for any nonlinearity f. Our approach enables us to state sharp qualitative results regarding uniqueness and ra-dial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. Lastly, we study in more detail the standard nonlinearities f(u) = u^p, p > 1.
In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-Δu+\fracμ{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), $$ with small initial data.\\ For $μ< \frac{N(q-1)}{2}$ and $μ\in (0, μ_*)$, where $μ_*>0$ is depending on the nonlinearties' powers and the space dimension ($μ_*$ satisfies $(q-1)\left((N+2μ_*-1)p-2\right) = 4$), we prove that the wave equation, in this case, behaves like the one without dissipation ($μ=0$). Our result…
We consider the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subc…
Global vs blow-up solutions and optimal threshold for hyperbolic ODEs with possibly singular nonlinearities
本文研究含可能奇异非线性项（含 MEMS 型）的双曲常微分方程，利用 Lyapunov 函数刻画小参数与小初值下的全局解及 omega 极限集，证明强扰动或大初值时爆破，并给出时变解与稳态解存在性的最优阈值。
Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities
本文研究了大型光滑星形区域及全空间上具有混合非线性项的 Kirchhoff 方程的给定质量解，分析了其存在性、多重性与渐近行为，并指出势函数的存在导致标准 Pohozaev 恒等式方法失效。
Refined blow-up behavior for semilinear heat equations with arbitrarily fast growing nonlinearities
该研究针对非尺度不变情形下含任意快速超指数增长非线性项的半线性热方程，推导了其径向非增正爆破解的精确全局爆破剖面，恢复了爆破剖面的普适结构。
Solitons, scattering and blow-up for the nonlinear Schrödinger equation with combined power-type nonlinearities on $\mathbb{R}^d\times\mathbb{T}$
本文研究波导流形上组合幂次非线性NLS的长时间动力学，通过构造两个重标度变分问题族，建立了孤立子、散射与爆破解的定性定量性质及周期性依赖结果。
We consider in this article the damped wave equation, in the \textit{scale-invariant case} with combined two nonlinearities, which reads as follows: \begin{displaymath} \d (E) \hspace{1cm} u_{tt}-Δu+\fracμ{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data.\\ Compared to our previous work \cite{Our}, we show in this article that the first hypothesis on the damping coefficient $μ$, namely $μ< \frac{N(q-1)}{2}$, can be removed, and the second one can be extended from $(0, μ_*/2)$ to $(0, μ_*)$ where $μ_*>0$ is solution of $(q-1)\left((N+μ_*-1)…
We are interested in this article in studying the damped wave equation with localized initial data, in the \textit{scale-invariant case} with mass term and two combined nonlinearities. More precisely, we consider the following equation: $$ (E) {1cm} u_{tt}-Δu+\fracμ{1+t}u_t+\frac{ν^2}{(1+t)^2}u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty), $$ with small initial data. Under some assumptions on the mass and damping coefficients, $ν$ and $μ>0$, respectively, we show that blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained in \cite{Ou…
Refined blow-up behavior for reaction-diffusion equations with non scale invariant exponential nonlinearities
本文研究了具非尺度不变指数非线性的半线性热方程的正径向递减爆破解，刻画了其渐近爆破行为并获得了原始变量尺度下精确的全局爆破剖面，揭示了由ODE预解式与固定时空构建块复合而成的普适爆破结构。
Asymptotic Blow-up Behavior for the Semilinear Heat Equation with Super-exponential Nonlinearities
本文研究超指数非线性半线性热方程径向I型爆破解的渐近行为，在维数不超过2时证明了爆破速率公式，通过变换将方程视为指数情形的扰动并估计附加项。
The focusing energy-critical nonlinear Schrödinger system with power-type growth nonlinearities in the radial case
本文在径向能量临界框架下研究一般幂型非线性聚焦薛定谔耦合系统，利用集中紧性与变分方法证明基态解存在，并建立散射与爆破二分性。
We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or la…
We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: \begin{align*} \begin{cases} u_{tt}- k(0) Δu - \int_0^{\infty} k'(s) Δu(t-s) ds + |u_t|^{m-1}u_t=|u|^{p-1}u, \;\;\;\;\; Ω\times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } Ω\times (-\infty,0], \end{cases} \end{align*} where $Ω$ is a bounded domain in $\mathbb R^3$ with a Dirichlét boundary condition. The relaxation kernel $k$ is monotone decreas…
Peakon solutions and analytical properties for the Camassa-Holm type equations with quadratic nonlinearities
本文研究了具有二次非线性的Camassa-Holm型方程的多峰解动力学系统及其初值问题的分析性质。首先在Besov空间中建立了局部well-posed性，并给出了破散准则。随后，通过适当的初始条件保证解要么全局存在要么在有限时间内破散。最后，利用非行波解证明了在Besov空间$B_{2,\infty}^{3/2}$中的ill-posed性。
The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly s…
Blow up and lifespan estimates for systems of semi-linear wave equations with dampings and potentials
本文研究了具有阻尼和势的半线性波动方程组，推导了其爆破区域和寿命估计。
A priori estimates of solutions to local and nonlocal superlinear parabolic problems
本文研究超线性抛物型问题变号解的先验估计，涵盖局部与非局部情形，应用包括爆破速率等，方法基于能量、插值及Pohozaev恒等式。
In this paper we develop a blow up analysis for solutions of a planar semilinear elliptic equation involving exponential nonlinearities. Such solutions describe cosmic strings, and we show how their blow up behaviour is characterised by new and surprising phenomena. For example in some cases, the blow up profile of the solution is described in terms of a differential problem that bares a geometrical meaning in the context of the uniformization of the Riemann sphere with conical singularities.
In this paper, we consider the semi-linear wave systems with power-nonlinearities and a large class of space-dependent damping and potential. We obtain the same blow-up regions and the lifespan estimates for three types wave systems, compared with the systems without damping and potential.
We study an elliptic problem with exponential nonlinearities describing the statistical mechanics equilibrium of point vortices with variable intensities. For suitable values of the physical parameters we exclude the existence of blow-up points on the boundary, we prove a mass quantization property and we apply our analysis to the construction of minimax solutions.
In this work we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms on the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will …
In this paper, we study the focusing nonlinear Schrödinger equation with exponential nonlinearities \[ i \partial_t u + Δu = - \left(e^{4π|u|^2} - 1 - 4πμ|u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $μ\in \{0, 1\}$. By using variational arguments, we first derive invariant sets where the global existence and finite time blow-up occur. In particular, we obtain sharp thresholds for global existence and finite time blow-up. In the case $μ=1$, by adapting a recent argument of Arora-Dodson-Murphy \cite{ADM}, we study the long time dynamics of g…
We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the $p-q$ plane. The main goal of this paper is to find out the sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of solutions.
We consider the focusing nonlinear Schrödinger equations $i\partial_t u+Δu +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up time
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.
We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of the fractional time derivatives, we present a critical value of the dimension N, for which global solutions with small data exist, otherwise solutions blow-up in finite time. Furthermore, the large time behavior of global solutions is discussed.
We investigate the Cauchy problem for a 2x2-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in R+ x RN. Under appropriate conditions on the exponents and the fractional orders of the time derivatives, it is shown that there exists a threshold value of the dimension N, for which, small data-global solutions as well as finite time blowing-up solutions exist. Furthermore, we investigate the L1-decay estimates of global solutions.
Blow-up phenomena ofvweakly coupled systems of several evolution equations, especially complex Ginzburg-Landau equationsvis shown by a straightforward ODE approach not so-called test-function method, which gives the natural blow-up rate. The difficulty of the proof is that, unlike the single case, terms which come from the fact that the Laplacian cannot be absorbed into the weakly coupled nonlinearities. A similar ODE approach is applied to heat systems by Mochizuki to obtain the lower estimate of lifespan.
Robin problem with measure data and singular nonlinearities on the boundary
本文研究带测度数据与边界奇异吸收项的Robin问题，证明正重整化解存在性，给出广义Feynman-Kac随机表示，并推导正则性与唯一性结果。
Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities
该研究针对带组合非线性项的双调和薛定谔方程，用新方法证明了归一化基态解的存在性、基态对应限制能量泛函的局部极小值，且驻波会因爆破强不稳定，结果涵盖临界情形。
Asymptotic blow-up behavior for the semilinear heat equation with non scale invariant nonlinearity
本文研究了非尺度不变非线性半线性热方程的渐近爆破行为，证明了一大类非线性项的正解在爆破点附近趋于对应 ODE 解，并给出了爆破集的紧致性等性质。
In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-Δu + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}μ(|s|)$. Here n is the space dimension and $μ$ is a modulus of continuity. Our goal is to obtain sharp conditions on $μ$ to obtain a threshold between global (in time) existence of small data solutions (stability of the zerosolution) and blow-up behavior even of small data solutions.
A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities. We start by performing a blow-up analysis in order to derive some information on the local blow-up masses. As a consequence we get a compactness property in a supercritical range. We next introduce a variational argument based on improved Moser-Trudinger inequalities which yields existence of solutions for any choice of the underlying su…
We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -Δ_1 u = g(u)|D u|+h(u)f & \text{in}\;Ω,\\ u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(Ω)$, and $g$ and $h$ are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities $g$ and $h$ produces remarkable regularizing effects on the solutions. The sharpness of our main results is discusse…
We study stable blow-up dynamics in the $L^2$-supercritical nonlinear Schrödinger equation in various dimensions. We first investigate the profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4] on the existence and local uniqueness of solutions of the cubic profile equation to other $L^2$-supercritical nonlinearities and dimensions $d \geq 2$. We then numerically observe the multi-bump structure of such solutions, and in particular, exhibit the $Q_{1,0}$ solution, a candidate for the stable blow-up profile. Next, using the dynamic rescaling method, we investigate stable …
We consider in this article the weakly coupled system of wave equations in the \textit{scale-invariant case} and with time-derivative nonlinearities. Under the usual assumption of small initial data, we obtain an improvement of the delimitation of the blow-up region by obtaining a new candidate for the critical curve. More precisely, we enhance the results obtained in \cite{Palmieri} for the system under consideration in the present work. We believe that our result is optimal in the sense that beyond the blow-up region obtained here we may conjecture the global existence of the solution.
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