Cauchy problem for evolution equations of higher order on \(t\) which contain a differential operator of infinite order (Q2768788)

Let for \(x\geq 0\), \(\Omega_{j}(x)=\int_{0}^{x}\omega_{j}(\xi) d\xi\), \(M_{j}(x)=\int_{0}^{x}\mu_{j}(\xi) d\xi\), \(\Omega_{j}(-x)=\Omega_{j}(x), M_{j}(-x)=M_{j}(x)\), where \(\omega_{j}:[0,+\infty)\to[0,+\infty)\), \(\omega_{j}(x),\;j=1,\ldots,n\) be continuous, increasing, \(\omega_{j}(0)=0, \omega_{j}(1)>1\), \(\lim_{x\to+\infty}\omega_{j}(x)=+\infty\), \(j=1,\ldots,n\); the functions \(\mu_{j}: [0,+\infty)\to[0,+\infty)\), \(j=1,\ldots,n \) have the same properties as the functions \(\omega_{j}(x)\). The space \(W_{M}^{\Omega}\equiv W_{M_1,\ldots,M_{n}}^{\Omega_1,\ldots,\Omega_{n}}\) is defined as a set of all entire functions \(\varphi: \mathbb C^n \to\mathbb C\) such that \(\exists C>0\), \(\exists a_{j}>0\), \(\exists b_{j}>0\), \(j=1,\ldots,n\), \(\forall z=x+iy\equiv(x_1+iy_2,\ldots,x_{n}+iy_{n})\in\mathbb C^n: |\varphi(z)|\equiv|\varphi(z_1,\ldots,z_{n})|\leq C\exp\{-M_1(a_1x_1)-\cdots-M_{n}(a_{n}x_{n})+\Omega_1(b_1y_1)+\cdots+\Omega_{n}(b_{n}y_{n})\}\). Let \(f(z)=\sum_{j=0}^{\infty}\sum_{|k|=j} c_{k}z_1^{k_1}\ldots z_{n}^{k_{n}}\) be some entire function. NEWLINENEWLINENEWLINEWe say that in the space \(W_{M}^{\Omega}\) the differential operator \(f(D)=\sum_{j=0}^{\infty}\sum_{|k|=j}c_{k}D^{k},\;D^{k}=\partial^{|k|}/\partial x_1^{k_1}\ldots\partial x_{n}^{k_{n}}\) is defined if for any test function \(\varphi\in W_{M}^{\Omega}\) the series NEWLINE\[NEWLINEf(D)\varphi(z)=\sum_{j=0}^{\infty}\sum_{|k|=j}c_{k}(D^{k}\varphi)(z)NEWLINE\]NEWLINE represents some test function from \(W_{M}^{\Omega}\). By \(P_{M}^{\Omega}\) we denote the set of entire single-valued functions \(\varphi: \mathbb C^{n}\to \mathbb C\) which are multiplicators in the space \(W_{M}^{\Omega}\) and \(e^{\varphi}\in W_{M}^{\Omega}\). NEWLINENEWLINENEWLINEThe author proves the correct solvability of the Cauchy problem NEWLINE\[NEWLINED_{t}^{\beta}u(t,x)=(-1)^{[\beta]+1}D_{t}^{\{\beta\}}B_{\varphi}u(t,x),\quad (t,x)\in(0,T]\times\mathbb{R}^{n},NEWLINE\]NEWLINE NEWLINE\[NEWLINED_{t}^{\{\beta\}}u(t,\cdot)|_{t=0}=f,NEWLINE\]NEWLINE where \(\varphi\in P_{M}^{\Omega}\); \(B_{\varphi}\) is a differential operator defined by function \(\varphi\); \(f\in (W_{M^1}^{\Omega^1}(\mathbb{R}^{n}))'\); \(\Omega^1\) and \(M^1\) are the functions dual to \(\Omega\) and \(M\), respectively; \((W_{M}^{\Omega})'\) is the space of linear continuous functionals on \(W_{M}^{\Omega}\).