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Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators. (English. Russian original) Zbl 1217.34100

Differ. Equ. 46, No. 8, 1164-1177 (2010); translation from Differ. Uravn. 46, No. 8, 1153-1166 (2010).
The authors are interested in the following Cauchy problem for an abstract Euler-Poisson-Darboux equation:
\[ u_{tt}+ A(t)u+{B(t)\over t}\;u_t= f(t),\quad t\in(0,T),\quad u(0)= 0,\quad u_t(0)= 0. \]
The families of operators \(\{A(t)\}\) and \(\{B(t)\}\) have a time-dependent domain \(D(A(t))\) and \(D(B(t))\) in a Hilbert space \(H\) for \(t\in(0, T)\). Both families satisfy standard assumptions (e.g., \(A(t)\) is self-adjoint and positive, \(B(t)\) is subordinate to \(\sqrt{A(t)}\) and so on). The authors are interested in well-posedness results, in particular in the determination of suitable function spaces for the right-hand side \(f= f(t)\) and for the solutions \(u= u(t)\). The solutions are generalized solutions (strong solutions) having energy generated by \(u_{tt}+ A(t)u\). Fixing the function space of the right-hand sides, the authors derive at first an energy estimate and show then that this energy inequality holds for all elements of this function space. This gives the desired well-posedness. Applications to mixed problems in 1D for hyperbolic equations with a singular coefficient are given.

MSC:

34G10 Linear differential equations in abstract spaces
35L20 Initial-boundary value problems for second-order hyperbolic equations
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[1] Gavrilova, N.V. and Yurchuk, N.I., The Cauchy Problem for Operator-Differential Equations of Euler-Poisson-Darboux Type, Differ. Uravn., 1981, vol. 17, no. 5, pp. 789–795. · Zbl 0472.34036
[2] Lomovtsev, F.E., Necessary and Sufficient Conditions for the Unique Solvability of the Cauchy Problem for Second-Order Hyperbolic Differential Equations with a Variable Domain of Operator Coefficients, Differ. Uravn., 1992, vol. 28, no. 5, pp. 873–886. · Zbl 0819.35088
[3] Lomovtsev, F.E., Second-Order Hyperbolic Operator-Differential Equations with Variable Domains of Smooth Operator Coefficients, Dokl. Nats. Akad. Nauk Belarusi, 2001, vol. 45, no. 1, pp. 34–37. · Zbl 1163.34371
[4] Lomovtsev, F.E., Boundary Value Problems for Differential-Operator Equations with Variable Domains of Smooth and Discontinuous Operator Coefficients, Doctoral (Phys.-Math.) Dissertation, Minsk, 2003.
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