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Solvability of a nonclassical parabolic problem arising in radiophysics. (English. Russian original) Zbl 0984.35070
Mosc. Univ. Math. Bull. 55, No. 5, 12-19 (2000); translation from Vestn. Mosk. Univ., Ser. I 2000, No. 5, 12-19 (2000).
In [Modelirovanie i analiz informatsionnykh sistem, Yaroslavl’ 3, 32-36 (1996)] V. F. Kambulov presented a model of parametric excitation of oscillations in the generator with RC-distributed parameters consisting in the periodic change of the capacity at the amplifier output. Moreover, the boundary value problem \[ \begin{aligned} u_t' = u_{xx}'',\quad u_x'|_{x=1} = 0,\quad [\alpha(t)u|_{x=0}]_t'+u|_{x=0} + F(u|_{x=1}) = 0,\\ \quad u|_{t_0=t} = u_0(x)\in W_2^1(0,1);\quad 0\leq x\leq 1,\quad t\geq t_0 \end{aligned}\tag{1} \] was considered, where \(\alpha(t)\) is a given function. The authors of this paper discuss the solvability of problem (1) under appropriate initial conditions.
35K15 Initial value problems for second-order parabolic equations
35B41 Attractors
35B15 Almost and pseudo-almost periodic solutions to PDEs