Hyperbolic Cauchy problems in a region with a characteristic boundary of full multiplicity (Q752316)

Let \(\Omega\) be a domain in \({\mathbb{R}}^ n\). The author considers the Cauchy problems in (0,T)\(\times \Omega\) for a class of hyperbolic differential operators, where (0,T)\(\times \partial \Omega\) is characteristic and the multiplicity of the characteristic is equal to the order of the operator. The Cauchy data are given on a non-characteristic hyperplane \(\{t=0\}\times \Omega\). The author proves existence and uniqueness theorems and corresponding estimates in Sobolev norms. Typical examples of the problems in consideration are the following: Find a solution u satisfying 1) \((\partial^ 2_ t-\rho (x)^ 2 \partial^ 2_ x-\partial^ 2_ y)u=f(t,x,y)\) in \(\{0<t<T\), \(x>0\), \(-\infty <y<\infty \}\) with initial conditions on \(\{t=0\), \(x>0\), \(-\infty <y<\infty \}\), where \(\rho (x)=x\) near \(x=0.\) 2) \((\partial^ 2_ t-\rho (x)^ 2 \partial^ 2_ x-\rho (y)^ 2 \partial^ 2_ y)u=f(t,x,y)\) in \(\{0<t<T\), \(x>0\), \(y>0\}\) with initial conditions on \(\{t=0\), \(x>0\), \(y>0\}.\) 3) \((\partial^ 2_ t-\rho (x,y)^ 2(\partial^ 2_ x+\partial^ 2_ y))u=f(t,x,y)\) in \(\{0<t<T\), (x,y)\(\neq (0,0)\}\) with initial conditions on \(\{t=0\), (x,y)\(\neq (0,0)\}\), where \(\rho (x,y)=(x^ 2+y^ 2)^{1/2}\) near \((x,y)=(0,0)\).