Properties of solutions of conjugacy problems for certain irregular equations (Q583476)

The author investigates the dependence of smoothness of solutions of a weakly irregular partial differential equation on the spectrum of differential operators contained in the equation. Let \(V=[-T_ 1,T_ 2]\times Q,\) \(T_ s>0\), \(s=1,2\), \(Q=\{x=(x_ 1,x_ 2,...,x_ n)\in {\mathbb{R}}^ n:\quad 0\leq x_ r\leq 2\pi,\quad r=1,2,...,n\},\) and \(V_ 1=[-T_ 1,0]\times Q,\) \(V_ 2=[0,T_ 2]\times Q.\) The equation considered in this note is \((1)\quad D_ t^ 3u-A(t,D_ x)u=f(t,x)\) in V, where \(A(t,D_ x)=A_ 1(d_ x)\) for \(t\in (-T_ 1,0)\); \(=A_ 2(D_ x)\) for \(t\in (0,T_ 2),\) with boundary conditions mentioned below. Let \(u=(u_ 1,u_ 2)\), where \(u_ 1\) and \(u_ 2\) are restrictions of u on \(V_ 1\) and \(V_ 2\), respectively. The boundary conditions imposed on u are periodic conditions: \[ (2)\quad u(t,x_ 1,...,x_ r+2\pi,...,x_ n)=u(t,x_ 1,...,x_ r,...,x_ n),\quad r=1,...,n; \] boundary conditions: \[ (3)\quad \sum^{2}_{p=0}(\beta_ p^{(m)}D_ t^ pu_ 1(-t_ 1,x)+\gamma_ p^{(m)}D_ t^ pu_ 2(T_ 2,x))=0,\quad m=1,2,3,\quad x\in Q; \] conjunction conditions on \(t=0:\) \[ (4)\quad D_ t^ pu_ 2(0,x)=\alpha_ pD_ t^ pu_ 1(0,x),\quad \alpha_ p>0,\quad p=0,1,2,\quad x\in Q. \] Let \(\lambda_{sk}\) (k\(\in ({\mathcal Z}_+)^ n\), \({\mathcal Z}_+={\mathcal N}\cup \{0\}\), \(s=1,2)\) be eigenvalues of \(A_ s\) the closure of \(A_ s(D_ x)\) with periodic conditions (2) in \(L_ 2(Q)\). One of the results in this note: Theorem 4. Let \(A_ 2(D_ x)=\theta A_ 1(D_ x)\), \(\theta >0\). Then under the conditions (2), (4) and boundary conditions: \[ (3')\quad u_ 1(-T_ 1,x)=u_ 2(T_ 2,x)=0,\quad \beta D_ tu_ 1(-T_ 1,x)+\gamma D_ tu_ x(T_ 2,x)=0, \] the unique solution u of (1) exists for each \(f\in L_ 2(V)\), and satisfies conditions \(D_ t^ 3u_ s\in L_ 2(V_ s)\), \(s=1,2\) if and only if there exist \(R>0\) and \(\phi\in (0,\pi /2)\) such that \(\arg (\lambda_{sk})\in [-\tau /2+\phi,\tau /2-\phi]\) follows from \(| \lambda_{sk}| \geq R\).