In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations

with \(p>1\), \(2\theta \in [0, \alpha ]\) and \(\delta \in (\theta ,\alpha ]\). We show that, under additional regularity \(\left( H^{\alpha +\delta }({\mathbb {R}}^n)\cap L^{m}({\mathbb {R}}^n) \right) \times \left( H^{2\delta }({\mathbb {R}}^n)\cap L^{m}({\mathbb {R}}^n)\right) \) for initial data, with \(m\in (1,2]\), the critical exponent is given by \(p_c=1+\frac{2m\theta }{n}\). The nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters \(\alpha , \delta , \theta \), by using the test function method (under suitable sign assumptions on the initial data).

在本文中，我们找到了半线性耗散发展方程的柯西问题的小数据解的整体指数（及时存在）的临界指数

通过\（p> 1 \），\（2 \ theta \ in [0，\ alpha] \）和\（\ delta \ in（\ theta，\ alpha] \）证明，在附加规则下\（ \ left（H ^ {\ alpha + \ delta}（{\ mathbb {R}} ^ n）\ cap L ^ {m}（{\ mathbb {R}} ^ n）\ right）\ times \ left（H ^ {2 \ delta}（{\ mathbb {R}} ^ n）\ cap L ^ {m}（{\ mathbb {R}} ^ n} \ right）\）用于初始数据，其中\（m \ in （1,2] \），关键指数由\（p_c = 1 + \ frac {2m \ theta} {n} \）给出，证明了在次临界情况下，对于整数，不存在全局解参数（（alpha，\ delta，\ theta \），通过使用测试函数方法（在初始数据的适当符号假设下）。