×

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)
PDF BibTeX XML Cite
Full Text: Link