Linearized stability for degenerate and singular semilinear and quasilinear parabolic problems: the linearized singular equation

Jesús Ildefonso Diaz, Jesús Hernández

DOI: http://dx.doi.org/10.12775/TMNA.2019.091

Abstract


We study some linear eigenvalue problems for the Laplacian operator with singular absorption or/and source coefficients arising in the linearization around positive solutions to some quasilinear degenerate parabolic equations and singular semilinear parabolic problems as well. We show that the linearization process applies even if the coefficients behave singularly with the distance to the boundary to the exponent two. This improves previous references in the literature. Applications to the above mentioned nonlinear problems are also presented.

Keywords


Linearization; linear eigenvalue problems with singular coefficients; quasilinear degenerate parabolic equations; singular semilinear parabolic problems

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References


G. Akagi and R. Kajikiya, Stability of stationary solutions for semilinear heat equations with concave nonlinearity, Commun. Contemp. Math. 17 (2015), no. 6, 1550001 1–1550001 29.

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

G. Anello and F. Faraci, Two solutions for an elliptic problem with two singular terms, Calc. Var. Partial Differential Equations 56 (2017), no. 4, 56–91.

D. Aronson, M.G. Crandall and L.A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion equation, Nonlinear Anal. 6 (1982), 1001–1022.

M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the pLaplace operator, Manuscripta Math. 109 (2002), 229–231.

S. Bensid and J.I. Dı́az., Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 5, 1757–1778.

S. Bensid and J.I. Dı́az, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete Contin. Dyn. Syst. Ser. B. 24 (2019), 1033–1047.

H. Berestycki, J.-P. Dias, M.J. Esteban and M. Figueira, Eigenvalue problems for some nonlinear Wheeler–Dewitt operators J. Math. Pures Appl. 72 (1993), 493–515.

M. Bertsch and R. Rostamian, The principle of linearized stability for a class of degenerate diffusion equations, J. Differential Equations 57 (1985), 373–405.

J. M. Bony, Principe du Maximum dans les espaces de Sobolev, C.R. Acad. Sci. Paris Ser. 265 (1967), 333–336.

B. Bougherara, J. Giacomoni and J. Hernández, Existence and regularity of weak solutions for singular elliptic equations Electron. J. Differ. Equ. Conf. 22 (2015), 19–30.

B. Bougherara, J. Giacomoni and S. Prashanth, Analytic global bifurcation and infinite turning points for very singular problems, Calc. Var. Partial Differential Equations 52 (2015), 829–856.

H. Brezis and M. Marcus, Hardy’s inequality revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), no. 4, 217–237.

H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000), 177–191.

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.

H. Brezis and L. Oswald, Remarks on sublinear problems, Nonlinear Anal. 10 (1986), 55–64.

X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. 156 (1998), no. 1, 30–56.

A.C. Casal and J.I. Dı́az, On the principle of pseudo-linearized stability: application to some delayed parabolic equations, Nonlinear Anal. 63 (2005), e997–e1007.

T. Cazenave, F. Dickstein and M. Escobedo, A semilinear heat equation with concave–convex nonlinearity, Rend. Mat. Appl. (7) 19 (1999), 211–242.

K.C. Chang, Nonlinear extensions of the Perron–Frobenius and the Krein–Routman theorems, J. Fixed Point Theory Appl. 15 (2014), 433–457.

M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with singular nonlinearity Comm. Partial Differential Equations 2 (1977), 193–222.

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Longman, Harlow, 1992.

A.N. Dao and J.I. Dı́az, A gradient estimate to a degenerate parabolic equation with a singular absorption term: global and local quenching phenomena, J. Math. Anal. Appl. 437 (2016), 445–473.

A.N. Dao, J.I. Dı́az and P. Sauvy, Quenching phenomenon of singular parabolic problems with L1 initial data, Electron. J. Differential Equations 136 (2016), 1–16.

J. Dávila and L. Dupaigne, Comparison results for PDEs with a singular potential, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 61–83.

J. Dávila and L. Dupaigne, Hardy type inequalities, J. Eur. Math. Soc. 6 (2004), no. 3, 335–365.

J. Dávila and M. Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math. 90 (2003), 303–335.

J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Trans. Amer. Math. Soc. 357 (2004), 1801–1828.

R. Dhanya, J. Giacomoni, S. Prashanth and K. Saoudi, Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in R2 , Adv. Differential Equations 17 (2012), no. 3–4 , 338-369.

J.I. Dı́az, On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via flat solutions: the one-dimensional case, Interfaces Free Bound. 17 (2015), 333–351.

J.I. Dı́az, On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case, SeMA J. 74 (2017), no. 3, 225–278; Correction to: On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case, SeMA J. 74 (2017), no. 3, 563–568.

J.I. Dı́az, Decaying to zero bifurcation solution curve for some sublinear elliptic eigenvalue type problems, Rev. Acad. Canaria Cienc. 29 (2019), 9–19.

J.I. Dı́az, D. Gómez-Castro and J.M. Rakotoson, Existence and uniqueness of solutions of Schrödinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications, Differ. Equ. Appl. 10 (2018), 47–74.

J.I. Dı́az, D. Gómez-Castro, J.M. Rakotoson and R. Temam, Linear equations with unbounded coefficients on the weighted space, Discrete Contin. Dyn. Syst. 38 (2018), no. 2, 509–546.

J.I. Dı́az, D. Gómez-Castro and J.L. Vázquez, The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach, Nonlinear Anal. (2018), 325–360.

J.I. Dı́az and J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C.R. Acad. Sci. Paris 329 (1999), 587–592.

J.I. Dı́az and J. Hernández, Positive and nodal solutions bifurcating from the infinity for a semilinear equation: solutions with compact support, Port. Math. 72 (2015), no. 2, 145–160.

J.I. Dı́az and J. Hernández, Linearized stability for degenerate and singular semilinear and quasilinear parabolic problems: the Lyapunov stability. (in preparation)

J.I. Dı́az, J. Hernández and Y. Il’yasov, On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption, Nonlinear Anal. 119 (2015), 484–500.

J.I. Dı́az, J. Hernández and Y. Il’yasov, Stability criteria on flat and compactly supported ground states of some non-Lipschitz autonomous semilinear equations, Chinese Ann. Math. 38 (2017), 345–378.

J.I. Dı́az, J. Hernández and Y. Il’yasov, Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ?3.

J.I. Dı́az, J. Hernández and F.J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems, J. Math. Anal. Appl. 352 (2009), 449–474.

J.I. Dı́az, J. Hernández and F.J. Mancebo, Nodal solutions bifurcating from infinity for some singular p-Laplace equations: flat and compact support solutions, Minimax Theory Appl. 2 (2017), 27–40.

J.I. Dı́az, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence term, Milan J. Maths. 79 (2011), 233–245.

J.I. Dı́az and J.E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilnéaire, C.R. Acad. Sci. Paris Sér. I Math. 305 (1987), 521–524.

J.I. Dı́az and I.I. Vrabie, Existence for reaction-diffusion systems. A compactness method approach, J. Math. Anal. Appl. 188 (1994), no. 2, 521–540.

J.I. Dı́az and I.I. Vrabie, On a Boussinesq type system in fluid dynamics, Topol. Methods Nonlinear Anal. 4 (1994), no. 2, 399–416.

P. Drábek and J. Hernández, Existence and uniqueness of positive solutions for some quasilinear elliptic problems, Nonlinear Anal. 44 (2001), 189–204.

P. Drábek and J. Hernández, Quasilinear eigenvalue problems with singular weights for the p-Laplacian, Ann. Mat. Pura Appl. 198 (2019), 1069–1086.

P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Berlin, De Gruyter, 1997.

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman and Hall/CRC, Boca Raton, FL, 2011.

N. El Berdan, J.I. Dı́az and J.M. Rakotoson, The uniform Hopf inequality for discontinuous coefficients and optimal regularity in bmo for singular problems, J. Math. Anal. Appl. 437 (2016), 350–379.

M. Fall and F. Mahmoudi, Weighted Hardy inequality with higher dimensional singularity on the boundary, Calc. Var. Partial Differential Equations 50 (2014), 779–798.

M. Fila, H.A.Levine and J.L.Vazquez, Stabilization of solutions of weakly singular quenching problems, Proc. Amer. Math. Soc. 119 (1993), 555–559.

J. Fleckinger, J. Hernández and F. de Thélin, Existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems, Boll. Unione Mat. Ital. 7-B (2004), no. 8, 159–188.

J. Fleckinger, J. Hernández, P. Takač and F. de Thélin, Uniqueness and positivity for solutions to equations with the p-Laplacian, Reaction-Diffusion Systems (G. Caristi and E. Mitidieri, eds.), Marcel Dekker, New York, 1998, pp. 141–155.

V. Galaktionov and J.L.Vazquez, Necessary and sufficient conditions of complete blowup and extinction for one-dimensional quasilinear heat equations, Arch. Ration. Mech. Anal. 129 (1995), 225–244.

M. Ghergu and V. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford University Press, 2008.

J. Giacomoni, P. Sauvy and S. Shmarev, Complete quenching for a quasilinear parabolic equation, J. Math. Anal. Appl. 410 (2014), 607–624.

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1977.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer–Verlag, New York, 1981.

J. Hernández and F.J. Mancebo, Singular elliptic and parabolic equations, Handbook of Differential Equations (M. Chipot and P. Quittner, eds.), Vol. 3, Elsevier, 2006, 317–400.

J. Hernández, F.J. Mancebo and J.M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 777–813.

J. Hernández, F. Mancebo and J.M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 41–62.

R. Kajikiya, Stability and instability of stationary solutions for sublinear parabolic equations, J. Differential Equations 264 (2018), 786–834.

O. Kavian, Inégalité de Hardy–Sobolev, C.R. Acad. Sci. Paris Sér. I 286 (1978), 779–781.

K. Kirschgässner and H. Kielhofer, Stability and bifurcation in fluid mechanics, Rocky Mountain Math. J. 3 (1973), 275–318.

M. Kucera, J. Necas and J. Soucek, The eigenvalue problem for variational inequalities and a new version of the Lusternik–Schnirelmann theory, Nonlinear Analysis (L. Cesari, ed.), Academic Press, New York, 1978, pp. 125–143.

A. Kufner, Weighted Sobolev Spaces, John Wiley and Sons, Inc. New York, 1985, (1980).

O.A. Ladyzhenskaya and V.A. Solonnikov, On a principle of linearization and invariant manifolds for problems of magnetic hydrodynamics. Boundary-value problems of mathematical physics and related problems of function theory, Part 7, Zap. Nauchn. Sem. Leningrad. Otdel. (LOMI) 38 (1973), 46–93 (in Russian); English transl.: J. Soviet Math. 8 (1977), 384–422.

V. Kh. Le and K. Schmitt, Global Bifurcation in Variational Inequalities, Springer, New York, 1997.

Y.Ch. Li, Linear hydrodynamic stability, Notices Amer. Math. Soc. 65 (2018), no. 10, 1255–1259.

P.L. Lions, Structure of the set of steady-state solutions and asymptotic behaviour of semilinear heat equations, J. Differential Equations 53 (1984), 362–384.

M. Marcus, V.J. Mizel and Y. Pinchover, On the best constant for Hardy’s inequality in RN , Trans. Amer. Math. Soc. 350 (1998), 3237–3255.

B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math., Vol. 279, Longman Scientific & Technical, Tiarlow, 1990.

D. Phillips, Existence of solutions of quenching problems, Appl. Anal. 24 (1987), 253–264.

A. Porretta, A note on the bifurcation of solutions for an elliptic sublinear problem, Rend. Semin. Mat. Univ. Padova 107 (2002), 153–164.

D.H. Sattinger, Stability of nonlinear parabolic systems, J. Math. Anal. Appl. 24 (1968), 241–245.

D.H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer–Verlag, New York, 1973.

P. Takač, Stabilization of positive solutions for analytic gradient-like sytems, Discrete Contin. Dyn. Syst. 6 (2000), 947–973.

I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd ed., Pitman Monographs, Longman, 1995.


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