【作者】 张卓；

【导师】 曹春玲；

【作者基本信息】 吉林大学 ， 应用数学， 2018， 硕士

【摘要】 本文讨论下列具有齐次Dirichlet边界条件的问题通过构造一个新的控制函数,建立(?)|u|p(x,t)dx与初始能量之间的关系,证明上述问题在初始能量为正时,解在有限时间内爆破,特别地,这里仅假定pt(x,t)是可积的,而不是一致有界的.这种弱条件在变指数情况下很少见到.此外,还建立了爆破时间的下界.主要结论如下:定理 1.假设指数 p(x,t)和 m(x,t)满足 1<p-≤ p(x,t)≤ p+<∞,1<m-≤m(x,t)≤ m+<∞,且下述条件成立则存在一个正数T,使得问题(0.1)有唯一的局部解u满足u ∈C[[0,T);H01(Ω))∩ Lp-(O,T;Lp(x,t)(Ω)),ut ∈C([0,T);L2(Ω))∩ Lm(Ω ×(0,T)).定理2.假设定理1中(H2)和(H3)以及下述条件成立(H6)存在一个足够小的常数0<ε0<1,使得1-ε0≤k<1.则爆破时间T*满足如下估计其中系数定义为其中B是嵌入H01(Ω)→ Lp+(Ω)的嵌入常数.更多还原

【Abstract】 This paper deals with the following problemBy constructing a new control function,the relationship between the term(?)|u|p(x,t)dx and the initial energy is established,and then we prove that the solution of the above problem blows up in a finite time for a positive initial energy,especially,it is only assumed that pr(x,t)is integrable rather than uniformly bounded.This weak condition is rarely seen in the case of variable exponents.Furthermore,a lower bound for the blow-up time is established.The main conclusions are as follows:Theorem 1.Suppose that the exponent p(x,t),m(x,t)satisfies 1<p-≤ p(x,t)≤p+<∞,1<m-≤m(x,t)≤ m+<∞,and the following conditions holdThen Problem(0.1)has a unique local solution u ∈C([0,T);H01(Ω))∩ Lp-(O,T;Lp(x,t)(Ω)),ut ∈C([0,T);L2(Ω))∩ Lm(Ω ×(0,T)),for some T.Theorem 2.Assume that(H2)and(H3)of Theorem 1 hold,and that the following conditions are satisfied:(H6)there exists a sufficiently small 0<ε0<1 such that 1-ε0≤k<1.Then the blow-up time T*satisfies the following estimate where the coefficients are defined by where B is the embedding constant with H01(Ω)→ Lp+(Ω).更多还原