A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

By Karen Yagdjian. 2004

78 citations recorded by the source corpus.

Abstract Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator y ∂ x 2 + ∂ y 2 on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator. We give for the Tricomi-type operator ∂ t 2 - t m △ x a fundamental solution relative to an arbitrary point of R n + 1 with the support in the region t ⩾ 0 , where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation . The application of that fundamental solution to the L p - L q estimate for the forced Tricomi-type equation is given as well.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/286a80d2929fabb2cda804d173407acc57aaf552

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In Applied Mathematics

1. A. Palmieri
2. Daoyin He
3. Michael Reissig
4. Ingo Witt
5. Huicheng Yin

In Mathematical Physics

1. A. Palmieri
2. Daoyin He
3. Michael Reissig
4. Ingo Witt
5. Huicheng Yin
6. M. DʼAbbicco
7. M. Ebert
8. Ryo Ikehata

Citations in the corpus

• On the global solution problem for semilinear generalized Tricomi equations, I (2015, background): “For instance, for the linea r generalized Tricomi equation, the authors in [1], [28] and [30] have computed its fundamental s olution explicitly.”
• A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass (2019, methodology): “ explicit integral representation formula for solutions to the inhomogeneous linear Cauchy problem in 1d related to (1.5). Here we strongly follow Yagdjian’s Integral Transform Approach of the papers [34, 38, 36, 37]. Finally, in Section 6 we prove Corollary 1.4. Some concluding remarks and open pro”
• Global Existence for the n-Dimensional Semilinear Tricomi-Type Equations (2006, unknown): “To find out that integral equation, we refer to the operator G introduced in (3.6) and (3.7) (Yagdjian, 2004).”