# Boundary value problems for twice degenerate differential equations with multiple characteristics

### Abstract

We study the solvability of boundary value problems for degenerate differential equations of the form $$\varphi(t)u_t-(-1)^m\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t)$$ ($D^k_x=\frac{\partial^k}{\partial x^k},\,m\geq 1$ is an integer, $x\in(0,1),\, t\in(0,T),\,0<T<+\infty),$ called equations with multiple characteristics. In these equations, the function $\varphi(t)$ can change the sign on the interval $[0, T]$ arbitrarily, while the function $\psi(t)$ is assumed nonnegative. For the equations under consideration, we propose the formulation of boundary value problems which are essentially determined by numbers $\varphi(0)$ and $\psi(T)$. Existence and uniqueness theorems are proved for the regular solutions that have all Sobolev generalized derivatives entering into the equation.

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*Mathematical notes of NEFU*, 25(4), pp. 34-44. doi: https://doi.org/10.25587/SVFU.2018.100.20552.

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