Existence and Uniqueness of Strong Solutions for a Class of Quasi-Linear Hyperbolic Equations with Order Degeneration
Rossitza I. Semerdjieva Β· Mathematische Nachrichten Β· 2002
Let k(y) > 0, π(y) > 0 for y > 0, k(0) = π(0) = 0 and limy β 0k(y)/π(y) exists; then the equation L(u) β k(y)uxx β βy(π(y)uy) + a(x, y)ux = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|AC = 0, where G is a simply connected domain in β2 with piecewise smooth boundary βG = ABβͺACβͺBC; AB = {(x, 0) : 0 β€ x β€ 1}, AC : x = F(y) = β«y0(k(t)/π(t))1/2dt and BC : x = 1 β F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies CarathΓ©odory condition and |f(x, y, u)| β€ Q(x, y) + b|u| with Q β L2(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u1) β f(x, y, u2| β€ C|u1 β u2|, where C = const > 0.