Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-03-23 , DOI: 10.1016/j.jde.2021.03.033 Marcello D'Abbicco
We study the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: where , and . Here either (singular problem) or (regular problem). We show that this model may be interpreted as a semilinear wave equation with borderline dissipation: the existence of global small data solutions depends not only on the power p, but also on the parameter μ. Global small data weak solutions exist if In the case of , the above condition is equivalent to , where is the critical exponent conjectured by W.A. Strauss for the semilinear wave equation without dissipation (i.e. ) in space dimension k. Varying the parameter μ, there is a continuous transition from (for ) to (for ). The optimality of follows by known nonexistence counterpart results for (and for any if ).
As a corollary of our result, we obtain analogous results for generalized semilinear Tricomi equations and other models related to the Euler-Poisson-Darboux equation.
中文翻译:
具有功率非线性的Euler-Poisson-Darboux方程的小数据解决方案
我们研究具有幂非线性的Euler-Poisson-Darboux方程的柯西问题: 在哪里 , 和 。在这里 (单个问题)或 (常规问题)。我们表明,该模型可以解释为具有边界耗散的半线性波动方程:全局小数据解的存在不仅取决于幂p,而且取决于参数μ。如果存在全球小数据弱解决方案 如果是 ,上述条件等同于 , 在哪里 是WA Strauss猜想的无耗散半线性波动方程的临界指数(即 )在空间尺寸k中。改变参数μ,从 (为了 ) 到 (为了 )。的最优性 接着是已知的不存在的对应结果 (以及任何 如果 )。
作为结果的推论,我们获得了广义半线性Tricomi方程和与Euler-Poisson-Darboux方程有关的其他模型的相似结果。