Mixed problem with an integral space variable condition for a third order parabolic equation of mixed type.(English)Zbl 1149.35394

Summary: We study a mixed problem with an integral space variable condition for a parabolic equation of mixed type
${\mathcal L}u= \frac{\partial^2u}{\partial t^2}- \biggl(\frac{\partial^3u}{\partial x^2\partial t}+ \frac1x \frac{\partial^2u}{\partial x\partial t}\biggr)= f(t,x) \quad\text{in }\Omega=(0,T)\times(0,1)$
initial conditions,
$lu= u(0,x)= \varphi(x), \quad x\in(0,1), \qquad l_1u= \frac{\partial u}{\partial t}(0,x)= \psi(x),\quad x\in(0,1),$
the Dirichlet condition $$u(t,1)=0$$, $$t\in (0,T)$$, and the integral condition
$\int_l^1u(t,\xi)\,d\xi=0, \quad 0<l<1,\;T\in(0,T).$
The existence and uniqueness of the solution in a functional weighted Sobolev space are proved. The proof is based on two sided a priori estimates and the density of the range of the operator generated by the considered problem.

MSC:

 35M10 PDEs of mixed type 35B45 A priori estimates in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35K70 Ultraparabolic equations, pseudoparabolic equations, etc. 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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