## Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations

Yang Hui,, Han Yuzhu,

School of Mathematics, Jilin University, Changchun 130012

 基金资助: 国家自然科学基金.  11401252吉林省教育厅基金.  JJKH20190018KJ

 Fund supported: the NSFC.  11401252the Education Department of Jilin Province.  JJKH20190018KJ

Abstract

In this paper, blow-up properties of solutions to an initial-boundary value problem for a parabolic type Kirchhoff equation are studied. The main results contain two parts. In the first part, we consider this problem with a general diffusion coefficient $M(\|\nabla u\|_2^2)$ and general nonlinearity $f(u)$. A new finite time blow-up criterion is established, and the upper and lower bounds for the blow-up time are also derived. In the second part, we deal with the case that $M(\|\nabla u\|_2^2)=a+b\|\nabla u\|_2^2$ and $f(u)=|u|^{q-1}u$, which was considered in[Computers and Mathematics with Applications, 2018, 75:3283-3297] with $q>3$, where global existence and finite time blow-up of solutions were obtained for subcritical, critical and supercritical initial energy. Their results are complemented in this paper in the sense that $q=3$ will be shown to be critical for the existence of finite time blow-up solutions to this problem.

Keywords： Kirchhoff equation ; General nonlinearity ; Blow-up ; Blow-up time ; Critical exponent

Yang Hui, Han Yuzhu. Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations. Acta Mathematica Scientia[J], 2021, 41(5): 1333-1346 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} u_t-M( \int_\Omega |\nabla u|^{2} {\rm d}x)\Delta u = k(t)f(u), \qquad &x\in\Omega, 0<t<T, \\ u(x, t) = 0, \qquad &x\in\partial\Omega, 0<t<T, \\ u(x, 0) = u_0(x), \qquad &x\in\Omega, \end{array}\right. \end{eqnarray}$

(H1)    $M\in C[0, \infty)$并且$M(t)\geq m_0>0$对任意$t\geq0$都成立, 其中$m_0$为正常数. 此外, 存在常数$\sigma\in(0, 1)$使得

$\begin{eqnarray} \overline{M}(t)\geq\sigma t M(t), \ \ \forall\ t\in {{\Bbb R}} ^+, \end{eqnarray}$

(H2)    $sf(s)\geq0, \quad \forall s\in {{\Bbb R}} ;$

(H3)    $f\in C^1({{\Bbb R}} )$并且存在常数$\alpha>\frac{2}{\sigma}-1$使得

(H4)    存在正整数$l$及常数$a_i>0\ (1\leq i\leq l)$使得

$$$\ J(u;t) = \frac{1}{2}\overline{M}\left(\|\nabla u\|_2^2\right)-k(t)\int_\Omega F(u) {\rm d}x, \qquad u\in H_0^1(\Omega), \ t\geq 0,$$$

$$$I(u;t) = M\left(\|\nabla u\|_2^2\right)\|\nabla u\|_2^2-k(t)\int_\Omega u f(u){\rm d}x, \qquad u\in H_0^1(\Omega), \ t\geq 0,$$$

$\begin{eqnarray} (u_t, \phi)+ M(\|\nabla u\|_2^2)(\nabla u, \nabla \phi) = (k(t)f(u), \phi), \quad{\rm a.e.}\ t\in(0, T). \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{t\rightarrow T}\|u(\cdot, t)\|_2^2 = +\infty. \end{eqnarray}$

$$$\frac{1}{S_{q+1}} = \inf\limits_{0\neq u\in H_{0}^{1}(\Omega)}\frac{\|\nabla u\|_2}{\|u\|_{q+1}}, \quad u\in H_0^1(\Omega).$$$

$q = 3$, $S_{q+1} = S_4$将被简写为$S$.

## 3 问题(1.1)的有限时刻爆破

$\rm(i) $$J(u_0; 0)<0 ; \rm(ii)$$ 0\leq J(u_0; 0)< m_0\lambda_1\left(\frac{\sigma}{2}-\frac{1}{\alpha+1}\right)\|u_0\|_2^2\triangleq C_0\|u_0\|_2^2$, 其中由(H1) 和(H3)可知$C_0>0$.

(i) 为研究情形(i), 我们利用一阶微分不等式方法, 该方法选自文献[18]. 设

$\begin{eqnarray} u f(u)\geq (\alpha+1)F(u), \ \ \alpha>\frac{2}{\sigma}-1. \end{eqnarray}$

$\begin{eqnarray} L'(t) = -I(u(t);t)&\geq&-\frac{1}{\sigma}\overline{M}(\|\nabla u\|_2^2)-(\alpha+1)J(u(t);t)+\\\frac{\alpha+1}{2}\overline{M}(\|\nabla u\|_2^2) &\geq& (\alpha+1)H(t). \end{eqnarray}$

$\begin{eqnarray} && F''(t)F(t)-\frac{\alpha+1}{2}(F'(t))^2 \\ & = &F''(t)F(t)-2(\alpha+1)\left(\int_0^t(u, u_\tau){\rm d}\tau+\beta(t+\sigma) \right)^2 \\ & = &F''(t)F(t)+2(\alpha+1)\left[Q(t)-\left(F(t)-(T^*-t)\|u_0\|_2^2\right) \left(\int_0^t\|u_\tau\|_2^2{\rm d}\tau+\beta\right) \right] \\ &\geq & F''(t)F(t)-2(\alpha+1)F(t)\left(\int_0^t\|u_\tau\|_2^2{\rm d}\tau+\beta\right) \\ &\geq& F(t)\bigg[-2M(\|\nabla u\|_2^2)\|\nabla u\|_2^2+(\alpha+1)\overline{M}(\|\nabla u\|_2^2)-2(\alpha+1)J(u_0;0) \\ &&+ 2(\alpha+1)\int_0^t\|u_\tau\|_2^2{\rm d}\tau+2\beta- 2(\alpha+1)\int_0^t\|u_\tau\|_2^2{\rm d}\tau- 2(\alpha+1)\beta \bigg] \\ &\geq & F(t)\bigg[\big[(\alpha+1)\sigma-2\big]m_0\lambda_1\|u(t)\|_2^2-2(\alpha+1)J(u_0;0)-2\alpha\beta \bigg]\\ &\geq & F(t)\bigg[\big[(\alpha+1)\sigma-2\big]m_0\lambda_1\|u_0\|_2^2-2(\alpha+1)J(u_0;0)-2\alpha\beta \bigg]\\ & = &2(\alpha+1)F(t)\left[C_0\|u_0\|_2^2-J(u_0;0)-\frac{\alpha\beta}{\alpha+1} \right]\geq0, \end{eqnarray}$

$\begin{eqnarray} L'(t) = -I(u;t)& = &-M(\|\nabla u\|_2^2)\|\nabla u\|_2^2 +k(t)\int_\Omega u f(u){\rm d}x\\ & \leq&\ k(t)\sum\limits_{i = 1}^{l}\int_\Omega a_i |u|^{p_i+1}{\rm d}x, \ \ \ t\in [0, T). \end{eqnarray}$

$\begin{eqnarray} \int_\Omega a_i |u|^{p_i+1}{\rm d}x\leq \frac{p_i+1}{p_l+1}\|u\|_{p_l+1}^{p_l+1}+\frac{(a_i)^{\frac{p_l+1}{p_l-p_i}}(p_l-p_i)}{p_l+1}|\Omega|, \ \ \ t\in [0, T). \end{eqnarray}$

$\begin{eqnarray} L'(t)\leq \widehat{C}2^{\gamma} L^{\gamma}(t), \ \ t\in[0, T), \end{eqnarray}$

## 4 问题(1.6) 的临界指数

$\rm(i) $$1<q<\min\{3, 2^{*}-1\} ; \rm(ii)$$ q = 3<2^{*}-1 $$b\geq S^{4} . 此外, 当 \rm(i) 成立时 其中 C^{*} = C^{*}(q, b, S_{q+1})>0 . \rm(ii) 成立时 (i) 在(2.3) 式中取 \phi = u , 根据 H_0^1(\Omega)\hookrightarrow L^{q+1}(\Omega) 并应用Young不等式, 我们有 \begin{eqnarray} \frac{1}{2}\frac{\rm d}{{\rm d}t}\|u\|_2^2+a\|\nabla u\|_2^2+b\|\nabla u\|_2^4 = \|u\|_{q+1}^{q+1} \leq S_{q+1}^{q+1}\|\nabla u\|_2^{q+1} \leq\frac{q+1}{4}b\|\nabla u\|_2^4+C^{*}, \end{eqnarray} 这意味着 \begin{eqnarray} \frac{1}{2}\frac{\rm d}{{\rm d}t}\|u\|_2^2+a\lambda_{1}\|u\|_2^2\leq C^{*}, \end{eqnarray} 其中 C^{*} = \frac{3-q}{4}\left(\frac{S_{q+1}^{q+1}}{b^{\frac{q+1}{4}}}\right)^{\frac{4}{3-q}} . 直接计算可得 这说明 u(x, t) 是问题(1.6) 的一个全局弱解. (ii) 根据(4.1) 式和 b\geq S^{4} , 很明显 通过解上述常微分不等式, 我们得到 定理4.1证毕. 为了说明 q = 3 的情形下问题(1.6) 既有全局解又有有限时刻爆破解, 下面的引理是至关重要的, 它断言当 b 适当小时, {\cal N} 是非空的. 引理 4.1 设 q = 3<2^{*}-1$$ b<S^{4}$.${\cal N}\ne\emptyset$.

因为$\Omega $${{\Bbb R}} ^{n} 中的有界光滑区域并且 q = 3<2^{*}-1 , 众所周知(2.9) 式中定义得常数是可以取到的, 即存在 \overline{u}\in H_0^1(\Omega)\backslash\{0\} 使得 \|\overline{u}\|_{4} = S\|\nabla \overline{u}\|_{2} . \widetilde{\lambda} = \Big[\frac{a}{(S^4-b)\|\nabla\overline{u}\|_2^2}\Big]^{1/2}>0 . 直接计算可知 这意味着 \widetilde{\lambda}\overline{u}\in{\cal N} . 因此, {\cal N}\ne\emptyset . 一旦 {\cal N} 非空, 如下三个引理可以通过文献[11] 中当 3<q<2^*-1 时的相似理论来推导, 因此具体证明略去. 引理 4.2 令 q = 3<2^*-1$$ b<S^{4}$. 则位势井$W$的井深$d$是正的.

$\rm(ii) $$I(u_{0})<0 , 则对任意 \delta_{1}<1<\delta_{2} 和任意 0<t<T , 都有 u(x, t)\in V_\delta . 引理 4.4 令 q = 3<2^*-1 , b<S^{4} , u\in H_0^1(\Omega)$$ r(\delta) = \frac{\sqrt{\delta a}}{S^2}$. 则有

$\rm(i) $$0\leq \|\nabla u\|_2\leq r(\delta) , 则 I_\delta(u)\geq0 . \rm(ii)$$ I_\delta(u)<0$, 则$\|\nabla u\|_2>r(\delta)$.

$\rm(iii) $$I_\delta(u) = 0 , 则 \|\nabla u\|_2 = 0$$ \|\nabla u\|_2\geq r(\delta)$.

假设$u(x, t)$全局存在, 则对任意$t\geq0$, $G(t) = \int_0^t\|u\|_2^2{\rm d}\tau$是良定义的. 取导数可得

$\begin{eqnarray} G^{\prime}(t) = \|u\|_2^2, \end{eqnarray}$

$\begin{eqnarray} G^{\prime\prime}(t) = 2(u_t, u) = -2\left(a\|\nabla u\|_2^2+b\|\nabla u\|_2^4-\|u\|_4^4\right) = -2I(u). \end{eqnarray}$

$\begin{eqnarray} J(u) = \frac{a}{4}\|\nabla u\|_2^2+\frac{1}{4}I(u). \end{eqnarray}$

$\begin{eqnarray} G^{\prime\prime}(t)G(t)-2\left(G^{\prime}(t)\right)^2 \geq 2a\lambda_1G^{\prime}(t)G(t)-8J(u_0)G(t)-4\|u_0\|_2^2G^{\prime}(t). \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{ll} u_t-\Delta u = |u|^{q-1}u, \qquad &x\in\Omega, 0<t<T, \\ u(x, t) = 0, \qquad &x\in\partial\Omega, 0<t<T, \\ u(x, 0) = u_0(x), \qquad &x\in\Omega \end{array}\right. \end{eqnarray}$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Brezis H . Functional Analysis. Sobolev Spaces and Partial Differential Equations. New York: Springer, 2010

Chipot M , Valente V , Caffarelli G V .

Remarks on a nonlocal problem involving the Dirichlet energy

Rend Semin Mat U Pad, 2003, 110, 199- 220

Chipot M , Savitska T .

Nonlocal p-Laplace equations depending on the $L.p$ norm of the Gradient

Adv Differential Equ, 2014, 19, 997- 1020

D'Ancona P , Shibata Y .

On global solvability of non-linear viscoelastic equation in the analytic category

Math Methods Appl Sci, 1994, 17, 477- 489

D'Ancona P , Spagnolo S .

Global solvability for the degenerate Kirchhoff equation with real analytic data

Invent Math, 1992, 108, 247- 262

Fu Y , Xiang M .

Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent

Appl Anal, 2016, 95, 524- 544

Gazzola F , Weth T .

Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level

Differ Integral Equ, 2005, 18, 961- 990

Ghisi M , Gobbino M .

Hyperbolic-parabolic singular perturbation for middly degenerate Kirchhoff equations: time-decay estimates

J Differ Equ, 2008, 245, 2979- 3007

Han Y .

Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity

Appl Math Lett, 2020, 99, 1- 7

Han Y , Gao W , Sun Z , Li H .

Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy

Comput Math Appl, 2018, 76, 2477- 2483

Han Y , Li Q .

Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy

Comput Math Appl, 2018, 75, 3283- 3297

Levine H A .

Some nonexistence and stability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+F(u)$

Arch Ration Mech Anal, 1973, 51, 371- 386

Li J , Han Y .

Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation

Math Model Anal, 2019, 24, 195- 217

Liao M , Gao W .

Blow-up phenomena for a nonlocal p-Laplace equation with Neumann boundary conditions

Arch Math, 2017, 108, 313- 324

Liu Y , Zhao J .

On potential wells and applications to semilinear hyperbolic equations and parabolic equations

Nonlinear Anal TMA, 2006, 64, 2665- 2687

Nishihara K .

On a global solution of some quasilinear hyperbolic equation

Tokyo J Math, 1984, 7, 437- 459

Payne L E , Sattinger D H .

Saddle points and instability of nonlinear hyperbolic equtions

Israel J Math, 1975, 22, 273- 303

Philippin G A , Proytcheva V .

Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems

Math Methods Appl Sci, 2006, 29, 297- 307

Xu R .

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

Math Comput Simulat, 2009, 80, 808- 813

Xu R , Su J .

Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations

J Funct Anal, 2013, 264, 2732- 2763

Zheng S , MChipot M .

Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms

Asymptotic Anal, 2005, 45, 301- 312

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