Nonlinear Evolution Equations with Infinitely Many Derivatives

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We study the nonlinear evolutionary euclidean bosonic string equation $$\begin{aligned} u_t = \Delta e^{-c \Delta }\,u + U(t,u) , \quad c > 0 \; \end{aligned}$$ on the Euclidean space \({\mathbb {R}}^n\). We interpret the nonlocal operator \(\Delta e^{-c\,\Delta }\) using entire vectors of \(\Delta \) in \(L^2({\mathbb {R}}^n)\). We prove that it generates a bounded holomorphic \(C_0\)-semigroup on \(L^2({\mathbb {R}}^n)\) (so that it also satisfies maximal \(L^p\) regularity) and we show the well-posedness of the corresponding nonlinear Cauchy problem.