W. Balser, Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke, Pacific J. Math, vol.188, issue.1, pp.53-63, 1999.

W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations, 2000.

W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations, vol.201, issue.1, pp.63-74, 2004.

W. Balser and M. Loday-richaud, Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables, Adv. Dyn. Syst. Appl, vol.4, issue.2, pp.159-177, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00364722

W. Balser and M. Miyake, Summability of formal solutions of certain partial differential equations, Acta Sci. Math. (Szeged), vol.65, issue.3-4, pp.543-551, 1999.

W. Balser and M. Yoshino, Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients, Funkcial. Ekvac, vol.53, pp.411-434, 2010.

B. L. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble), vol.42, issue.3, pp.517-540, 1992.

M. Canalis-durand, J. Ramis, R. Schäfke, and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math, vol.518, pp.95-129, 2000.

O. Costin, H. Park, and Y. Takei, Borel summability of the heat equation with variable coefficients, J. Differential Equations, vol.252, issue.4, pp.3076-3092, 2012.

T. Gramchev and G. Lysik, Uniform analytic-Gevrey regularity of solutions to a semilinear heat equation, vol.81, pp.213-226, 2008.

M. Hibino, Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I & II, J. Differential Equations, vol.227, issue.2, pp.499-563, 2006.

M. Hibino, On the summability of divergent power series solutions for certain first-order linear PDEs, Opuscula Math, vol.35, issue.5, pp.595-624, 2015.

P. Hilton and J. Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer, vol.13, issue.2, pp.64-75, 1991.

K. Ichinobe, On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients, J. Differential Equations, vol.257, issue.8, pp.3048-3070, 2014.

D. A. Klarner, Correspondences between plane trees and binary sequences, J. Combinatorial Theory, vol.9, pp.401-411, 1970.

A. Lastra and S. Malek, On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, J. Differential Equations, vol.259, pp.5220-5270, 2015.

A. Lastra and S. Malek, On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems, Adv. Differ. Equ, 2015.

A. Lastra, S. Malek, and J. Sanz, On Gevrey solutions of threefold singular nonlinear partial differential equations, J. Differential Equations, vol.255, pp.3205-3232, 2013.

A. Lastra and H. Tahara, Maillet type theorem for nonlinear totally characteristic partial differential equations, Math. Ann, 2019.

D. A. Lutz, M. Miyake, and R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J, vol.154, pp.1-29, 1999.

G. Lysik and S. Michalik, Formal solutions of semilinear heat equations, J. Math. Anal. Appl, vol.341, pp.372-385, 2008.

S. Malek, On the summability of formal solutions of linear partial differential equations, J. Dyn. Control Syst, vol.11, issue.3, pp.389-403, 2005.

S. Malek, On the summability of formal solutions of nonlinear partial differential equations with shrinkings, J. Dyn. Control Syst, vol.13, issue.1, pp.1-13, 2007.

S. Malek, On the Stokes phenomenon for holomorphic solutions of integrodifferential equations with irregular singularity, J. Dyn. Control Syst, vol.14, issue.3, pp.371-408, 2008.

S. Malek, On Gevrey functions solutions of partial differential equations with fuchsian and irregular singularities, J. Dyn. Control Syst, vol.15, issue.2, pp.277-305, 2009.

S. Malek, On Gevrey asymptotic for some nonlinear integro-differential equations, J. Dyn. Control Syst, vol.16, issue.3, pp.377-406, 2010.

S. Malek, On the summability of formal solutions for doubly singular nonlinear partial differential equations, J. Dyn. Control Syst, vol.18, issue.1, pp.45-82, 2012.

S. Michalik, Summability of divergent solutions of the n-dimensional heat equation, J. Differential Equations, vol.229, pp.353-366, 2006.

S. Michalik, Summability of formal solutions to the n-dimensional inhomogeneous heat equation, J. Math. Anal. Appl, vol.347, pp.323-332, 2008.

S. Michalik, On the multisummability of divergent solutions of linear partial differential equations with constant coefficients, J. Differential Equations, vol.249, pp.551-570, 2010.

S. Michalik, Summability and fractional linear partial differential equations, J. Dyn. Control Syst, vol.16, issue.4, pp.557-584, 2010.

S. Michalik, Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients, J. Dyn. Control Syst, vol.18, issue.1, pp.103-133, 2012.

S. Michalik, On the multisummability of divergent solutions of linear partial differential equations with constant coefficients, vol.249, p.551, 2010.

, J. Differential Equations, vol.255, pp.2400-2401, 2013.

M. Miyake, Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations, J. Math. Soc. Japan, vol.43, issue.2, pp.305-330, 1991.

M. Miyake, Borel summability of divergent solutions of the Cauchy problem to nonKovaleskian equations, Partial differential equations and their applications, pp.225-239, 1999.

M. Miyake and Y. Hashimoto, Newton polygons and Gevrey indices for linear partial differential operators, Nagoya Math. J, vol.128, pp.15-47, 1992.

M. Miyake and A. Shirai, Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain, Ann. Polon. Math, vol.74, pp.215-228, 2000.

M. Miyake and A. Shirai, Structure of formal solutions of nonlinear first order singular partial differential equations in complex domain, Funkcial. Ekvac, vol.48, pp.113-136, 2005.

M. Miyake and A. Shirai, Two proofs for the convergence of formal solutions of singular first order nonlinear partial differential equations in complex domain, vol.37, pp.137-151, 2013.

M. Nagumo, Über das Anfangswertproblem partieller Differentialgleichungen, Jap. J. Math, vol.18, pp.41-47, 1942.

S. Ouchi, Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, vol.2, pp.375-417, 1995.

S. Ouchi, Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, vol.185, issue.2, pp.513-549, 2002.

S. Ouchi, Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields, J. Math. Soc. Japan, vol.57, issue.2, pp.415-460, 2005.

M. E. Pli? and B. Ziemian, Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables, Ann. Polon. Math, vol.67, issue.1, pp.31-41, 1997.

G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, vol.I, 1954.

J. Ramis, Théorèmes d'indices Gevrey pour leséquations différentielles ordinaires, Mem. Amer. Math. Soc, vol.48, p.95, 1984.

J. Ramis and Y. Sibuya, Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type, Asymptotic Anal, vol.2, issue.1, pp.39-94, 1989.

P. Remy, Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients
URL : https://hal.archives-ouvertes.fr/hal-02117418

P. Remy, Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients, J. Dyn. Control Syst, vol.22, issue.4, pp.693-711, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01171006

P. Remy, Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients, J. Dyn. Control Syst, vol.23, issue.4, pp.853-878, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01267736

P. Remy, Gevrey properties and summability of formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems, J. Dyn. Control Syst
URL : https://hal.archives-ouvertes.fr/hal-01778574

A. Shirai, Maillet type theorem for nonlinear partial differential equations and newton polygons, J. Math. Soc. Japan, vol.53, pp.565-587, 2001.

A. Shirai, Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac, vol.45, pp.187-208, 2002.

A. Shirai, A maillet type theorem for first order singular nonlinear partial differential equations, Publ. RIMS. Kyoto Univ, vol.39, pp.275-296, 2003.

A. Shirai, Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, vol.1431, pp.94-106, 2005.

A. Shirai, Alternative proof for the convergence or formal solutions of singular first order nonlinear partial differential equations, Journal of the School of Education, vol.1, pp.91-102, 2008.

A. Shirai, Gevrey order of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Journal of the School of Education, vol.6, pp.159-172, 2013.

A. Shirai, Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, part ii, Opuscula Math, vol.35, issue.5, pp.689-712, 2015.

H. Tahara and H. Yamazawa, Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations, vol.255, issue.10, pp.3592-3637, 2013.

W. Walter, An elementary proof of the Cauchy-Kowalevsky theorem, Amer. Math. Monthly, vol.92, issue.2, pp.115-126, 1985.

A. Yonemura, Newton polygons and formal Gevrey classes, Publ. Res. Inst. Math. Sci, vol.26, pp.197-204, 1990.