Problem With Shift for a "Pointwise" Loaded Hyperbolic-Parabolic Equation

Extension of the Tricomi problem for a loaded parabolic–hyperbolic equation with a characteristic line of change of type

“…Proof of Theorem The uniqueness of the solution to the problem $$$ {T}_1 $$$ for even $$$ n $$$, with regard to maximum principle for a loaded hyperbolic–parabolic equation, 30 follows from the fact that in the case of homogeneity, the problem ()–() has only the trivial solution. Consequently, it is sufficient to show that the solution $$$ u\left(x,y\right) $$$ of the Bitsadze problem is identically zero for $$$ {\varphi}_1(y)\equiv {\varphi}_2(y)\equiv {\psi}_k(x)\equiv 0 $$$, at $$$ k=1,2,\dots, n $$$.…”

“…$$ Hence, taking into account Abdullayev 31 that $$$ {D}_{0x}^{\beta_i}\tau (x)>0 $$$, ( $$$ {D}_{0x}^{\beta_i}\tau (x)<0 $$$) for $$$ \tau (x)>0 $$$ ( $$$ \tau (x)<0 $$$), due to $$$ {\beta}_i<0,{\mu}_2>0 $$$, ( $$$ {\tau}^{\prime}\left({x}_0\right)>0 $$$), we get: $$$$ \nu \left({x}_0\right)\ge 0,\left(\nu \left({x}_0\right)\le 0\right). $$$$ However, we know 30 that for $$$ y>0 $$$ the positive maximum functions $$$ u\left(x,y\right) $$$ in $$$ \overline{\Omega} $$$ can be reached only by …”

“…Proof of Theorem 3.1. The uniqueness of the solution to the problem T 1 for even n, with regard to maximum principle for a loaded hyperbolic-parabolic equation, 30 follows from the fact that in the case of homogeneity, the problem (1.2)-(1.4) has only the trivial solution. Consequently, it is sufficient to show that the solution u(x, 𝑦) of the Bitsadze problem is identically zero for 𝜑 1 (𝑦) ≡ 𝜑 2 (𝑦) ≡ 𝜓 k (x) ≡ 0, at k = 1, 2, … , n.…”

“…However, we know 30 that for 𝑦 > 0 the positive maximum functions u(x, 𝑦) in Ω can be reached only by AA 0 , AB, BB 0 .…”