Sets of initial values of smooth solutions of evolution equations with a differential operator of infinite order (Q2761531)

Let \(\Omega(x)=\int_{0}^{x}\omega(\xi) d\xi\), \(M(x)=\int_{0}^{x}\mu(\xi) d\xi\), \(\Omega(-x)=\Omega(x)\), \(M(-x)=M(x)\) for \(x\geq 0\), where \(\omega:[0,+\infty)\to[0,+\infty)\), \(\omega(x)\) is continuous, increasing, \(\omega(0)=0\), \(\omega(1)>1\), \(\lim_{x\to+\infty}\omega(x)=+\infty\); the function \(\mu: [0,+\infty)\to[0,+\infty)\) has the same properties as the function \(\omega(x)\). The space \(W_{M}^{\Omega}\) is defined as a set of entire functions \(\phi: {\mathbb{C}}\to {\mathbb{C}}\) such that \(\exists C>0\), \(\exists a>0\), \(\exists b>0\), \(\forall z=x+iy\in {\mathbb{C}}: |\phi(z)|\leq C\exp\{-M(ax)+\Omega(by)\}\). Let \(f(z)=\sum_{n=0}^{\infty}c_{n}z^{n}\) be an entire function. We say that in the space \(W_{M}^{\Omega}\) the differential operator \(f(D)=\sum_{n=0}^{\infty}c_{n}D^{n}\), \(D=d/dz\) is given if for any test function \(\phi\in W_{M}^{\Omega}\) the series \(f(D)\phi(z)=\sum_{n=0}^{\infty}c_{n}(D^{n}\phi)(z)\) represents some test function from \(W_{M}^{\Omega}\). By \(P_{M}^{\Omega}\) we denote the set of entire single-valued functions \(\phi: {\mathbb{C}}\to {\mathbb{C}}\) which are multipliers in the space \(W_{M}^{\Omega}\) and \(e^{\phi}\in W_{M}^{\Omega}\). The authors find the smooth solutions of the Cauchy problem NEWLINE\[NEWLINE{\partial u\over\partial t}=B_{\phi}(u),\;(t,x)\in(0,T]\times R;\;u(t,\cdot)|_{t=0}=f,NEWLINE\]NEWLINE where \(\phi\in P_{M}^{\Omega}\); \(B_{\phi}\) is a differential operator defined by the function \(\phi\); \(f\in (W_{M_1}^{\Omega_1})'\); \(\Omega_1\) and \(M_1\) are functions dual to \(\Omega\) and \(M\), respectively; \((W_{M}^{\Omega})'\) is the space of linear continuous functionals on \(W_{M}^{\Omega}\). An estimate of the derivatives of the fundamental solution is obtained.