On the Semilinear Heat Equations and Wave Equations with Unbounded Coefficient

We consider first initial-boundary value problems to semilinear heat equation(1) uₜ-uₓₓ+frac1rho(x)mathrmuq=f(mathrmx, t) , and semilinear wave equationwhere $-infty<a<x<b<infty$ , $t>0$ , q is a positive integer, f and g are given smooth functions in (x, mathrmi) and $rho(x)$ is a smooth function in x $in(a, b)$ and it has zero pointsUnder suitable conditions on $rho(x)$ , we first show the local existence of classical solutions to (1), (2), and in the second, we find sufficient conditions for obtaining global existence.$(a<xi<x)$, this implies |v(x)|leqq(x-a)|vᵖʳⁱᵐᵉ|inftyleqq Const.(x-a)|v|₂ , since $a<xleqq(a+b)/2$, we get $2(b-x)/(b-a)geqq 1$ , thus we obtainAnalogously, we also have |v(x)|leqqfrac2(x-a)(b-x)b-a Const.|v|₂ , fraca+b2leqq x<b .This shows our assertion.