zbMATH — the first resource for mathematics

On weak and strong solution operators for evolution equations coming from quadratic operators. (English) Zbl 06852558
Summary: We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on \(L^2(\mathbb{R}^n)\) which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behavior with the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.
The article generalizes the authors’ results in [Int. Math. Res. Not. 2015, No. 17, 8275–8288 (2015; Zbl 1339.47056)].

35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35K30 Initial value problems for higher-order parabolic equations
47A45 Canonical models for contractions and nonselfadjoint linear operators
47D06 One-parameter semigroups and linear evolution equations
Zbl 1339.47056
Full Text: DOI arXiv