非线性边界条件下拟线性瞬态方程组的Phragmén-Lindel?f型二择一结果

Phragmén-Lindel?f type alternative results for quasilinear transient equations with nonlinear boundary conditions

考虑了一类定义在三维半无穷柱体上的拟线性方程组,其中假设方程的解在柱体的有限端和侧面满足非齐次条件。定义了“能量”表达式,通过限制非线性项,利用微分不等式技术,推导了一阶微分不等式,解此不等式得到二择一结果,即证明了“能量”随与有限端距离的增大要么呈指数式(多项式)增加,要么呈指数式(多项式)衰减。同时,在衰减情形下得到了全能量的上界。

In this paper, we consider a class of quasilinear equations defined on a three-dimensional semi-infinite cylinder, in which the solutions of the equations are assumed to satisfy the nonhomogeneous conditions on both the finite end and the side of the cylinder. An expression of "energy" is defined. By limiting the nonlinear terms and making full use of the differential inequality technique, we obtain a first order differential inequality. By solving this inequality,we prove the alternative results, i. e., the "energy" increases exponentially(polynomial) or decays exponentially(polynomial) with the distance from the finite end. Finally, in the case of decay, we get the upper bound of total energy.