Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results

Abstract

For the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray-Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence. # 2002 Elsevier Science (USA)

Critical exponents for semilinear equations of mixed elliptic‐hyperbolic and degenerate types

“…As for weak solutions in the mixed-type case, the authors have established existence of nontrivial solutions for the case of the Tricomi operator for classes of nonlinearities with subcritical growth [15] in spaces of generalized solutions W 1 AC∪σ ;1 ( ), which is the closure of C…”

The self-similar solutions of the Tricomi-type equations

“…The Fourier transform of ϕ(x) = |x| −a is Φ(ξ) = C a,n |ξ| a−n , where according to [6] and to (14) of Sec 6.561 [8] one has…”

“…[2], [34], [36] for the bibliography.) Recently the linear and the semilinear Tricomi equations became the focus of interest of many authors (see, also [2], [3], [14], [15], [24], [25], [26], and [27]). …”

Nonexistence for mixed-type equations with critical exponent nonlinearity in a ball

“…Tricomi problems and mixed-type equations like (1.1) with f (u) = 0 or f (u) = λu and other types have been widely considered (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). The existence and uniqueness were obtained in these earlier papers under suitable conditions on different domains.…”