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A strong solution of a high-order mixed type partial differential equation with integral conditions. (English) Zbl 1160.35055

In the rectangle \(Q= (0,1)\times (0,T)\) the following equation is considered:
\[ {\partial^2u\over\partial t^2}+ (-1)^\alpha a(t){\partial^{2\alpha+1}u\over\partial x^{2\alpha}\partial t}= f(x,t), \tag{1} \]
where \(0< a_0< a(t)\leq a_1\) and \(0< a_2\leq (da(t)/dt)\leq a_3\) for \(t\in (0,T)\) and \(\alpha\in \mathbb{N}^*\). To equation (1) the following conditions are added:
a)
the initial conditions \(u(x, 0)= \varphi(x)\), \({\partial u\over\partial t} (x, 0)= \psi(x)\), \(x\in (0,1)\),
b)
the boundary conditions
\[ \begin{aligned} {\partial^i\over\partial x^i} u(0, t)&= 0\quad\text{for }0\leq i\leq\alpha- k,\quad t\in (0,T),\\ {\partial^i\over\partial x^i} (1,t)&= 0\quad\text{for }0\leq i\leq\alpha- k-1,\quad t\in (0,T),\end{aligned} \]
and
c)
the integral conditions
\[ \int^1_0 x^i u(\xi, t)\,d\xi= 0\quad\text{for }0\leq i\leq 2k-2,\quad 1\leq k\leq\alpha,\quad t\in (0,T), \]
where \(\varphi\), \(\psi\) are known functions, which satisfy compability conditions.
The existence and uniqueness of a strong solution are proved. The proof is based on energy inequality and on 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
35G10 Initial value problems for linear higher-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:

[1] Bouziani A, Kobe J. Math. 15 pp 47– (1998)
[2] Cannon JR, Quart. Appl. Math. 21 pp 155– (1963)
[3] Ionkin NI, Differ. Uravn. 13 pp 294– (1977)
[4] Kamynin NI, U.S.S.R. Comput. Math. and Math. Phys. 4 pp 33– (1964) · Zbl 0206.39801
[5] Choi YS, Nonlinear Anal. 18 pp 317– (1992) · Zbl 0757.35031
[6] DOI: 10.1016/0309-1708(91)90055-S
[7] Shi P, Theoretical Aspects of Industrial Design pp 76– (1992)
[8] Samarski AA, Differ. Uravn. 16 pp 1221– (1980)
[9] Batten GW, Math. Comp. 17 pp 405– (1963)
[10] Beilin AB, Electron. J. Diff. Equa. 2001 pp 1– (2001)
[11] Benouar NE, Differ. Equ. 27 pp 1482– (1991)
[12] DOI: 10.1137/0732025 · Zbl 0831.65094
[13] Cannon JR, Encyclopedia of Mathematics and its Applications 23 (1984)
[14] Kartynnik AV, Differ. Equ. 26 pp 1160– (1990)
[15] DOI: 10.1137/0524004 · Zbl 0810.35033
[16] Yurchuk NI, Differ. Equ. 22 pp 1457– (1986)
[17] Pulkina LS, Electron. J. Diff. Eqns. 1999 pp 1– (1999)
[18] Volkodavov VF, Differ. Equ. 34 pp 501– (1998)
[19] Denche M, Appl. Math. Lett. 13 pp 85– (2000) · Zbl 0956.35072
[20] Electron. J. Differential. Equations 2000 pp 1– (2000)
[21] Int. J. Math. Math. Sci. 26 pp 417– (2001) · Zbl 1005.35004
[22] DOI: 10.1155/S1048953303000054 · Zbl 1035.35085
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