A priori estimates of solutions to two problems for the generalized parabolic-hyperbolic heat equation (Q1363931)

Let the domain \(D\) be the union of the triangle with vertices at the points \(A(0,0)\), \(B(1,0)\), and \(C(1/2,1/2)\); the rectangle with vertices at the points \(A\), \(B\), \(M(0,-1)\), and \(N(1,-1)\); and the interval \(AB\). Consider the following equation of mixed parabolic-hyperbolic type on the set \(D^+\cup D^-\): \[ K(y)u_{xx}- u_{yy}+ a(x,y)u_x+ b(x,y)u_y+ c(x,y)u= f(x,y),\tag{1} \] where \(K(y)= [1+\text{sgn}(y)]/2\). We consider solutions \(u(x,y)\) such that there exist two functions, \(u^+(x,y)\in C^1(D^+\cup AB)\) and \(u^-(x,y)\in C^1(D^-\cup AB\cup MN)\), coinciding with \(u(x,y)\) on the sets \(D^+\) and \(D^-\), respectively, and satisfying the condition \[ {\partial u^-(0,x)\over\partial y}= \lambda\int^x_0 {\partial u^+(0,t)\over\partial y} e^{-\mu(x-t)}dt,\quad x\in(0,1),\tag{2} \] where \(\lambda\) and \(\mu\) are fixed positive numbers. This paper concerns two boundary value problems for (1), (2).