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Nonlinear integroparabolic equations on unbounded domains: existence of classical solutions with special properties. (Russian, English) Zbl 0977.35065
Sib. Mat. Zh. 42, No. 3, 585-609 (2001); translation in Sib. Math. J. 42, No. 3, 495-516 (2001).
Under consideration is the following parabolic equation: \[ \frac{\partial \rho}{\partial t}= \frac{\partial^2 \rho}{\partial \omega^2} + \varepsilon \frac{\partial^2 \rho}{\partial \theta^2} - F(\omega) \frac{\partial \rho}{\partial \theta} + \frac{\partial }{\partial \omega}(F\rho)- \Omega \frac{\partial \rho}{\partial \omega} -K_s(\theta,t) \frac{\partial \rho}{\partial \omega},\;\tag{1} \] where \((\theta,\omega,t,\Omega)\in Q=[0,2\pi]\times {\mathbb R} \times [0,T]\times [-G,G]\), \(\varepsilon>0\), \[ K_s(\theta,t)= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{2\pi} g(\Omega)\sin(\varphi-\theta)\rho(\varphi,\omega,t,\Omega) d\varphi d\omega d\Omega, \] and \(F(\omega)\) is a given function of a Hölder class. Equation (1) is a parabolic regularization of the nonlinear ultraparabolic equation of the Fokker-Plank type which arises in physics. The equation is furnished with the following boundary and initial conditions: \[ (\rho,\rho_{\theta})|_{\theta=0}=(\rho,\rho_{\theta})|_{\theta=2\pi},\;\;\rho|_{t=0}=\rho_0(\theta,\omega,\Omega). \tag{2} \] Using a priori bounds and the method of successive approximations, the authors establish the existence and uniqueness theorem for classical solutions to problem (1), (2), i.e., a solution belongs to some Hölder space in \(Q\). Moreover, under the corresponding conditions on the initial data, it is shown that a constructed solution is exponentially decreasing in \(\omega\), nonnegative, and the equality \[ \int_0^{2\pi}\int_{-\infty}^{\infty} \rho(\theta,\omega,t,\Omega) d\omega d\theta=1 \] holds whenever \[ \int_0^{2\pi}\int_{-\infty}^{\infty}\rho_0(\theta,\omega,\Omega) d\omega d\theta=1 . \]
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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