Nonlinear generalized Dunkl-wave equations and applications
A class of nonlinear generalized Dunkl-wave equations \[ \square_{k,\beta}u(t,x)=F(u(t,x)),\quad (t,x)\in [0,T]\times\mathbb R^{d+1}_+,\tag \(*\) \] where \(\square_{k,\beta}=-\partial^2_t+\Delta_{k,\beta}\), \(\Delta_{k,\beta}\) is the Dunkl-Bessel operator, and \(F\) is a nonlinear function, is introduced. The Dunkl-Bessel operator is defined as \(\Delta_{k,\beta}= \Delta_{k}+L_\beta\), where \[ L_\beta=\frac{d^2}{dx^2_{d+1}}+\frac{2\beta+1}{x_{d+1}}\frac{d}{dx_{d+1}},\quad \beta>-\frac12, \] and \(\Delta_{k}=\sum_{j=1}^dT^2_j\) (\(T_j\) is the Dunkl operator) is the Dunkl-Laplace operator on \(\mathbb R^d\). This paper continues investigations of the Cauchy problem for the Dunkl-wave equation started in [\textit{H. Mejjaoli}, J. Math. Anal. Appl. 346, No. 1, 41--54 (2008; Zbl 1149.35017)]. For the above equation \((*)\) the following problems are considered: 1) existence and uniqueness of solutions of the Cauchy problem; 2) approximation in the energy space of sequence solutions for the nonlinear generalized Dunkl-wave equation \(\square_{k, \beta}u_n=F(u_n)\) in \(\mathbb R\times\mathbb R^{d+1}_+\), with uniformly compactly supported Cauchy data converging weakly to \(H^1_{k,\beta}(\mathbb R^{d+1}_+)\times L^2_{k,\beta}(\mathbb R^{d+1}_+)\), with the sequence solutions of the linear generalized Dunkl-wave equation with the same Cauchy data. The proof uses Strichartz type inequalities and the energy estimate.
scientific article; zbMATH DE number 5835284
scientific article