Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities

"In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, $$a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f_{p\left( .\right)}\left( u\right)$$ in which $f_{p\left( .\right)}\left( u\right)$ the source term, $a(x,t)>0$ is a nonnegative function, and the exponents of nonlinearity $m(x)$, $p(x)$ are given measurable functions. Under suitable conditions on the given data, a finite-time blow-up result of the solution is shown if the initial datum possesses suitable positive energy, and in this case, we precise estimate for the lifespan $T^{\ast }$ of the solution. A blow-up of the solution with negative initial energy is also established."

Spin(7) Instantons and Hermitian Yang–Mills Connections for the Stenzel Metric

We use the highly symmetric Stenzel Calabi–Yau structure on $$T^{\star }S^{4}$$ T ⋆ S 4 as a testing ground for the relationship between the Spin(7) instanton and Hermitian–Yang–Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with structure group SO(3) and construct the moduli space of invariant Spin(7) instantons in this setting. This is comprised of two 1-parameter families—one of them explicit—of irreducible Spin(7) instantons. Each carries a unique HYM connection. We thus negatively resolve the question regarding the equivalence of the two gauge theoretic PDEs. The HYM connections play a role in the compactification of this moduli space, exhibiting a removable singularity phenomenon that we aim to further examine in future work.

Existence and nonexistence of global solutions for logarithmic hyperbolic equation

In this paper, we study the initial-boundary value problem of a class of degenerate quasilinear hyperbolic equation with logarithmic nonlinearity. By applying Galerkin method and the logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. Meanwhile,the global nonexistence of solutions is verified by means of the concavity analysis when the initial energy is positive and appropriately bounded.

Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

<p style='text-indent:20px;'>In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.</p>

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

<p style='text-indent:20px;'>In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document}</tex-math></inline-formula> is a nondecreasing and continuous function, <inline-formula><tex-math id="M2">\begin{document}$ [u]_{\alpha, 2} $\end{document}</tex-math></inline-formula> is the Gagliardo-seminorm of <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> are the fractional Laplace operators, <inline-formula><tex-math id="M6">\begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document}</tex-math></inline-formula> is a positive nonincreasing function and <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.</p>

Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources

<p style='text-indent:20px;'>The aim of this paper is to give global nonexistence and blow–up results for the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{tt}-\Delta u+P(x,u_t) = f(x,u) \qquad &amp;\text{in $(0,\infty)\times\Omega$,}\\ u = 0 &amp;\text{on $(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t) = g(x,u)\qquad &amp;\text{on $(0,\infty)\times \Gamma_1$,}\\ u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x) &amp; \text{in $\overline{\Omega}$,} \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded open <inline-formula><tex-math id="M2">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> subset of <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Gamma = \partial\Omega $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ (\Gamma_0,\Gamma_1) $\end{document}</tex-math></inline-formula> is a partition of <inline-formula><tex-math id="M7">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \Gamma_1\not = \emptyset $\end{document}</tex-math></inline-formula> being relatively open in <inline-formula><tex-math id="M9">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \Delta_\Gamma $\end{document}</tex-math></inline-formula> denotes the Laplace–Beltrami operator on <inline-formula><tex-math id="M11">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \nu $\end{document}</tex-math></inline-formula> is the outward normal to <inline-formula><tex-math id="M13">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, and the terms <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ Q $\end{document}</tex-math></inline-formula> represent nonlinear damping terms, while <inline-formula><tex-math id="M16">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ g $\end{document}</tex-math></inline-formula> are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.</p>