中文摘要：

考虑一类具有变指数伪抛物型方程的第一初边值问题.对于一般光滑区域Ω,先通过Galerkin方法构造问题的逼近解,然后在参数满足一定条件下利用能量估计方法得到逼近解的一致性先验估计,进而证明该类问题弱解的存在唯一性.

英文摘要：

We studied the first initial and boundary value problems for a class of pseudo-parabolic equations with variable exponent in a smooth domain Ω. Firstly, we constructed the approximate solutions through Galerkin method, then under some conditions limited on the parameter, using energy method, we obtained the uniform estimates for the approximate solutions. At last, using the above estimate, we gave the existence and uniqueness result for the original problem.