[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic
Press, 1975.
[2] G. Akagi, K. Ishige and R. Sato, The Cauchy problem for the Finsler heat
equation, to appear.
[3] D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces
for a class of evolution equations with strongly nonlinear sources, Ann. Scuola
Norm. Sup. Pisa Cl. Sci. 18 (1991), 363–441.
[4] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann.
Scuola Norm. Sup. Pisa 22 (1968), 607–694.
[5] D. G. Aronson and L. A. Caffarelli, The initial trace of a solution to the porous
medium equation, Trans. Amer. Math. Soc. 280 (1983), 351–366.
[6] J. M. Arrieta, A. N. Carvalho and A. Rodr´ıguez-Bernal, Parabolic problems
with nonlinear boundary conditions and critical nonlinearities, J. Differential
Equations 156 (1999), 376–406.
[7] J. M. Arrieta and A. Rodr´ıguez-Bernal, Non well posedness of parabolic equations with supercritical nonlinearities, Commun. Contemp. Math. 6 (2004),
733–764.
[8] C. Bandle, H. A. Levine and Qi S. Zhang, Critical exponents of Fujita type
for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl. 251
(2000), 624–648.
[9] P. Baras and R. Kersner, Local and global solvability of a class of semilinear
parabolic equations, J. Differential Equations 68 (1987), 238–252.
[10] P. Baras and M. Pierre, Crit`ere d’existence de solutions positives pour des
´equations semi-lin´eaires non monotones, Ann. Inst. H. Poincar´e Anal. Non
Lin´eaire 2 (1985), 185–212.
24
[11] P. B´enilan, M. G. Crandall and M. Pierre, Solutions of the porous medium
equation in RN under optimal conditions on initial values, Indiana Univ. Math.
J. 33 (1984), 51–87.
[12] G. Bernard, Existence theorems for certain elliptic and parabolic semilinear
equations, J. Math. Anal. Appl. 210 (1997), 755–776.
[13] M. F. Bidaut-V´eron, E. Chasseigne and L. V´eron, Initial trace of solutions
of some quasilinear parabolic equations with absorption, J. Funct. Anal. 193
(2002), 140–205.
[14] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian
perturbed by gradient operators, Comm. Math. Phys. 271 (2007), 179–198.
[15] K. Bogdan, A. St´os and P. Sztonyk, Harnack inequality for stable processes on
d-sets, Studia Math. 158 (2003), 163–198.
[16] M. Bonforte, Y. Sire and J. L. V´azquez, Optimal existence and uniqueness
theory for the fractional heat equation, Nonlinear Anal. 153 (2017), 142–168.
[17] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,
J. Anal. Math. 68 (1996), 277–304.
[18] C. Bucur, Some observations on the Green function for the ball in the fractional
Laplace framework, Commun. Pure Appl. Anal. 15 (2016), 657–699.
[19] K. Deng, M. Fila and H. A. Levine, On critical exponents for a system of heat
equations coupled in the boundary conditions, Acta Math. Univ. Comenianae
63 (1994), 169–192.
[20] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), 83–118.
[21] E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces
for a degenerate parabolic equation, Trans. Amer. Math. Soc. 314 (1989), 187–
224.
[22] E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution pLaplacian equation. Initial traces and Cauchy problem when 1 < p < 2, Arch.
Rational Mech. Anal. 111 (1990), 225–290.
[23] F. Dickstein, Blowup stability of solutions of the nonlinear heat equation with
a large life span. J. Differ. Equ. 223, 303–328 (2006)
[24] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
25
[25] J. Fern´andez Bonder and J. D. Rossi, Life span for solutions of the heat equation
with a nonlinear boundary condition, Tsukuba J. Math. 25 (2001), 215–220.
[26] J. Filo and J. Kaˇcur, Local existence of general nonlinear parabolic systems.
Nonlinear Anal. 24 (1995), 1597–1618.
[27] A. Z. Fino and M. Kirane, Qualitative properties of solutions to a time-space
fractional evolution equation, Quart. Appl. Math. 70 (2012), 133–157.
[28] Y. Fujishima and N. Ioku, Existence and nonexistence of solutions for a heat
equation with a superlinear source term, J. Math. Pures Appl. 118 (2018),
128–158.
[29] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut =
∆u + u1+α , J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.
[30] V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat
equations with nonlinear flux conditions on the boundary, Israel J. Math. 94
(1996), 125–146.
[31] M. Guedda and M. Kirane, A note on nonexistence of global solutions to a
nonlinear integral equation, Bull. Belg. Math. Soc. Simon Stevin 6 (1999),
491–497.
[32] C. Gui and X. Wang, Life spans of solutions of the Cauchy problem for a
semilinear heat equation. J. Differ. Equ. 115 (1995), 166–172
[33] M. A. Herrero and M. Pierre, The Cauchy problem for ut = ∆um when 0 <
m < 1, Trans. Amer. Math. Soc. 291 (1985), 145–158.
[34] K. Hisa and K. Ishige, Existence of solutions for a fractional semilinear parabolic
equation with singular initial data, Nonlinear Anal. 175 (2018), 108–132.
[35] K. Hisa and K. Ishige, Solvability of the heat equation with a nonlinear boundary condition, SIAM J. Math. Anal. 51 (2019), 565–594.
[36] M. Ikeda and M. Sobajima, Sharp upper bound for lifespan of solutions to
some critical semilinear parabolic, dispersive and hyperbolic equations via a
test function method, Nonlinear Anal. 182 (2019), 57–74.
[37] K. Ishige, On the existence of solutions of the Cauchy problem for a doubly
nonlinear parabolic equation, SIAM J. Math. Anal. 27 (1996), 1235–1260.
[38] K. Ishige and T. Kawakami, Global solutions of the heat equation with a nonlinear boundary condition, Calc. Var. Partial Differential Equations 39 (2010)
429–457.
26
[39] K. Ishige, T. Kawakami and K. Kobayashi, Global solutions for a nonlinear
integral equation with a generalized heat kernel, Discrete Contin. Dyn. Syst.
Ser. S 7 (2014), 767–783.
[40] K. Ishige, T. Kawakami and S. Okabe, Existence of solutions for a higher-order
semilinear parabolic equation with singular initial data, arXiv:1909.05492v1.
[41] K. Ishige, T. Kawakami and M. Sier˙ze¸ga, Supersolutions for a class of nonlinear
parabolic systems, J. Differential Equations 260 (2016), 6084–6107.
[42] K. Ishige and J. Kinnunen, Initial trace for a doubly nonlinear parabolic equation, J. Evol. Equ. 11 (2011), 943–957.
[43] K. Ishige and R. Sato, Heat equation with a nonlinear boundary condition and
uniformly local Lr spaces, Discrete Contin. Dyn. Syst. 36 (2016), 2627–2652.
[44] K. Ishige and R. Sato, Heat equation with a nonlinear boundary condition and
growing initial data, Differential Integral Equations 30 (2017), no. 7-8, 481–504.
[45] N. Ju, The maximum principle and the global attractor for the dissipative 2D
quasi-geostrophic equations, Comm. Math. Phys. 255 (2005), 161–181.
[46] T. Kan and J. Takahashi, Time-dependent singularities in semilinear parabolic
equations: behavior at the singularities, J. Differential Equations 260 (2016),
7278–7319.
[47] T. Kan and J. Takahashi, Time-dependent singularities in semilinear parabolic
equations: existence of solutions, J. Differential Equations 263 (2017), 6384–
6426.
[48] A. G. Kartsatos and V. V. Kurta, On blow-up results for solutions of inhomogeneous evolution equations and inequalities, J. Math. Anal. Appl. 290 (2004),
76–85.
[49] M. Kirane and M. Qafsaoui, Global nonexistence for the Cauchy problem of
some nonlinear reaction–diffusion systems, J. Math. Anal. Appl. 268 (2002),
217–243.
[50] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes
equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959–1014.
[51] T.-Y. Lee, Some limit theorems for super-Brownian motion and semilinear differential equations. Ann. Probab. 21 (1993), 979–995.
27
[52] T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of
solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc.
333 (1992), 365–378.
[53] Y. Maekawa and Y. Terasawa, The Navier–Stokes equations with initial data in
uniformly local Lp spaces, Differential Integral Equations 19 (2006), 369–400.
[54] M. Marcus and L. V´eron, Initial trace of positive solutions of some nonlinear
parabolic equations, Comm. Partial Differential Equations 24 (1999), 1445–
1499.
[55] J. Matos and P. Souplet, Universal blow-up rates for a semilinear heat equation
and applications, Adv. Differential Equations 8 (2003), 615–639.
[56] N. G. Meyers, A theory of capacities for potentials of functions in Lebesgue
classes, Math. Scand. 26 (1970), 255–292.
[57] N. Mizoguchi and E. Yanagida, Blowup and life span of solutions for a semilinear
parabolic equation, SIAM J. Math. Anal. 29 (1998), no. 6, 1434–1446.
[58] N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data
in a semilinear parabolic equation, Indiana Univ. Math. J. 50 (2001), no. 1,
591–610.
[59] T. Ozawa and Y. Yamauchi, Life span of positive solutions for a semilinear
heat equation with general non-decaying initial data, J. Math. Anal. Appl. 379
(2011) 518–523.
[60] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global
Existence and Steady States, Birkh¨auser Advanced Texts: Basler Lehrb¨
ucher,
Birkh¨auser Verlag, Basel, 2007.
[61] J. C. Robinson and M. Sier˙ze, ga, Supersolutions for a class of semilinear heat
equations, Rev. Mat. Complut. 26 (2013), 341–360.
[62] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian:
regularity up to the boundary, J. Math. Pures Appl. 101 (2014), 275–302.
[63] S. Sato, Life span of solutions with large initial data for a superlinear heat
equation, J. Math. Anal. Appl. 343 (2008), 1061–1074.
[64] H. Shang and F. Li, Singular parabolic equations with measures as initial data,
J. Differential Equations 247 (2009), 1720–1745.
[65] H. Shang and F. Li, On the Cauchy problem for the evolution p-Laplacian
equations with gradient term and source and measures as initial data, Nonlinear
Anal. 72 (2010), 3396–3411.
28
[66] S. Sugitani, On nonexistence of global solutions for some nonlinear integral
equations, Osaka J. Math. 12 (1975), 45–51.
[67] J. Takahashi, Solvability of a semilinear parabolic equation with measures as
initial data, Geometric properties for parabolic and elliptic PDE’s, 257–276,
Springer Proc. Math. Stat., 176, Springer, 2016.
[68] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in Lp , Indiana Univ. Math. J. 29 (1980), 79–102.
[69] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear
heat equation, Israel J. Math. 38 (1981), 29–40.
[70] D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math.
Soc. 55 (1944), 85–95.
[71] M. Yamaguchi and Y. Yamauchi, Life span of positive solutions for a semilinear
heat equation with non-decaying initial data, Differential Integral Equations 23
(2010) 1151–1157.
[72] Y. Yamauchi, Life span of solutions for a semilinear heat equation with initial
data having positive limit inferior at infinity, Nonlinear Anal. 74 (2011), no.
15, 5008–5014.
[73] K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition, Classics in
Mathematics, Springer-Verlag, Berlin, 1995.
[74] X. Z. Zeng, The critical exponents for the quasi-linear parabolic equations with
inhomogeneous terms, J. Math. Anal. Appl. 332 (2007), 1408–1424.
[75] Qi S. Zhang, A new critical phenomenon for semilinear parabolic problems, J.
Math. Anal. Appl. 219 (1998), 125–139.
[76] Qi S. Zhang, Blow up and global existence of solutions to an inhomogeneous
parabolic system, J. Differential Equations 147 (1998), 155–183.
[77] Qi S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds,
Duke Math. J. 97 (1999), 515–539.
[78] J. Zhao, On the Cauchy problem and initial traces for the evolution p-Laplacian
equations with strongly nonlinear sources, J. Differential Equations 121 (1995),
329–383.
[79] J. Zhao and Z. Xu, Cauchy problem and initial traces for a doubly nonlinear
degenerate parabolic equation, Sci. China 39 (1996), 673–684.
29
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