# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. (English) Zbl 1218.35052

The problem

$\begin{array}{cccccc}\hfill {u}_{t}-{\Delta }u& ={|\nabla u|}^{p},\hfill & \hfill \phantom{\rule{1.em}{0ex}}& x\in {\Omega },\hfill & \hfill \phantom{\rule{1.em}{0ex}}& t>0,\hfill \\ \hfill u\left(x,t\right)& =0,\hfill & \hfill \phantom{\rule{1.em}{0ex}}& x\in \partial {\Omega },\hfill & \hfill \phantom{\rule{1.em}{0ex}}& t>0,\hfill \\ \hfill u\left(x,0\right)& ={u}_{0}\left(x\right),\hfill & \hfill \phantom{\rule{1.em}{0ex}}& x\in {\Omega },\hfill & & \hfill \end{array}$

where ${u}_{0}\in {X}_{+}=\left\{v\in {C}^{1}\left(\overline{{\Omega }}$); $v\ge 0$, ${v|}_{\partial {\Omega }}=0\right\}$, is considered. This problem admits a unique maximal, nonnegative classical solution $u\in {C}^{2,1}\left(\overline{{\Omega }}×\left(0,T\right)\right)\cap {C}^{1,0}\left(\overline{{\Omega }}×\left[0,T\right)\right)$, where $T=T\left({u}_{0}\right)$ is the maximal existence time and ${\parallel u\left(t\right)\parallel }_{\infty }\le {\parallel {u}_{0}\parallel }_{\infty }$, $0, by the maximum principle. Sine (1)–(3) is well-posed in ${X}_{+}$, it follows that, if $T<\infty$, then

$\underset{t\to T}{lim}{\parallel \nabla u\left(t\right)\parallel }_{\infty }=\infty ·$

This phenomenon of $\nabla u$ blowing up with $u$ remaining uniformly bounded is known as gradient blow-up. The question of the location of gradient blow-up points within the boundary for problem (1)–(3) with $n\ge 2$ has not been addressed so far. The gradient blow-up set of $u$ is defined by

$\text{GBUS}\left({u}_{0}\right)=\left\{{x}_{0}\in \partial {\Omega };\phantom{\rule{4pt}{0ex}}\nabla u\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{unbounded}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\left(\overline{{\Omega }}\cap {B}_{\eta }\left({x}_{0}\right)\right)×\left(T-\eta ,T\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{4.pt}{0ex}}\eta >0\right\}·$

Note that by definition, $\text{GBUS}\left({u}_{0}\right)$ is compact. The main goal of this paper is to show that under some assumptions on ${\Omega }\subset {ℝ}^{2}$ and ${u}_{0}$, the gradient blow-up set $\text{GBUS}\left({u}_{0}\right)$ contains only one point. A possible physical interpretation is that the surface tension (diffusion) forces the steep region to become more and more concentrated near a single boundary point.

##### MSC:
 35B44 Blow-up (PDE) 35K58 Semilinear parabolic equations 82C24 Interface problems (dynamic and non-equilibrium); diffusion-limited aggregation
##### References:
 [1] Alaa N.: Weak solutions of quasilinear parabolic equations with measures as initial data. Ann. Math. Blaise Pascal 3(2), 1–15 (1996) [2] Alikakos N.D., Bates P.W., Grant C.P.: Blow up for a diffusion-advection equation. Proc. Roy. Soc. Edinburgh Sect. A 113(3–4), 181–190 (1989) [3] Angenent S.B., Fila M.: Interior gradient blow-up in a semilinear parabolic equation. Diff. Int. Eq. 9(5), 865–877 (1996) [4] Arrieta J.M., Rodriguez-Bernal A., Souplet Ph.: Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3(1), 1–15 (2004) [5] Asai K., Ishimura N.: On the interior derivative blow-up for the curvature evolution of capillary surfaces. Proc. Amer. Math. Soc. 126(3), 835–840 (1998) · Zbl 0992.35015 · doi:10.1090/S0002-9939-98-04084-2 [6] Barles G., Da Lio F.: On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations. J. Math. Pures Appl. 83, 53–75 (2004) · Zbl 1056.35071 · doi:10.1016/S0021-7824(03)00070-9 [7] Bartier J.-Ph., Souplet Ph.: Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition. C. R. Acad. Sci. Paris Sér. I Math. 338, 533–538 (2004) [8] Benachour S., Dabuleanu S.: The mixed Cauchy-Dirichlet problem for a viscous Hamilton-Jacobi equation. Adv. Diff. Eq. 8(12), 1409–1452 (2003) [9] Benachour S., Dăbuleanu-Hapca S., Laurençot Ph.: Decay estimates for a viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions. Asymptot. Anal. 51(3-4), 209–229 (2007) [10] Benachour S., Karch G., Laurençot Ph.: Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations. J. Math. Pures Appl. 83, 1275–1308 (2004) · Zbl 1064.35075 · doi:10.1016/j.matpur.2004.03.002 [11] Ben-Artzi M., Souplet Ph., Weissler F.B.: The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces. J. Math. Pures Appl. (9) 81(4), 343–378 (2002) · Zbl 1046.35046 · doi:10.1016/S0021-7824(01)01243-0 [12] Conner G.R., Grant C.P.: Asymptotics of blowup for a convection-diffusion equation with conservation. Diff. Int. Eq. 9(4), 719–728 (1996) [13] Dłotko T.: Examples of parabolic problems with blowing-up derivatives. J. Math. Anal. Appl. 154(1), 226–237 (1991) · Zbl 0743.35007 · doi:10.1016/0022-247X(91)90082-B [14] Fila M., Lieberman G.M.: Derivative blow-up and beyond for quasilinear parabolic equations. Diff. Int. Eq. 7(3-4), 811–821 (1994) [15] Filippov A.: Conditions for the existence of a solution of a quasi-linear parabolic equation (Russian). Dokl. Akad. Nauk SSSR 141, 568–570 (1961) [16] Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood cliffs, NJ (1964) [17] Friedman A., McLeod B.: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34(2), 425–447 (1985) · Zbl 0576.35068 · doi:10.1512/iumj.1985.34.34025 [18] Gidas B., Ni W.-M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 [19] Giga Y.: Interior derivative blow-up for quasilinear parabolic equations. Discrete Contin. Dyn. Syst. 1(3), 449–461 (1995) · Zbl 0878.35015 · doi:10.3934/dcds.1995.1.449 [20] Giga Y., Kohn R.V.: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36(1), 1–40 (1987) · Zbl 0601.35052 · doi:10.1512/iumj.1987.36.36001 [21] Gilding B.H.: The Cauchy problem for u t = ${\Delta }$u + || q , large-time behaviour. J. Math. Pures Appl. (9) 84(6), 753–785 (2005) · Zbl 1100.35046 · doi:10.1016/j.matpur.2004.11.003 [22] Gilding B.H., Guedda M., Kersner R.: The Cauchy problem for u t = ${\Delta }$u + || q . J. Math. Anal. Appl. 284(2), 733–755 (2003) · Zbl 1041.35026 · doi:10.1016/S0022-247X(03)00395-0 [23] Guo J.-S., Hu B.: Blowup rate estimates for the heat equation with a nonlinear gradient source term. Disc. Cont. Dyn. Syst. 20(4), 927–937 (2008) · Zbl 1159.35009 · doi:10.3934/dcds.2008.20.927 [24] Halpin-Healy T., Zhang Y-C.: Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Phy. Re. 254, 215–414 (1995) · doi:10.1016/0370-1573(94)00087-J [25] Herrero M.A., Velázquez J.J.L.: Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 131–189 (1993) [26] Hesaaraki M., Moameni A.: Blow-up positive solutions for a family of nonlinear parabolic equations in general domain in ${ℝ}^{n}$ . Michigan Math. J. 52(2), 375–389 (2004) · Zbl 1072.35095 · doi:10.1307/mmj/1091112081 [27] Kardar M., Parisi G., Zhang Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986) · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889 [28] Krug J., Spohn H.: Universality classes for deterministic surface growth. Phys. Rev. A 38, 4271–4283 (1988) · doi:10.1103/PhysRevA.38.4271 [29] Kutev, N.: Gradient blow-ups and global solvability after the blow-up time for nonlinear parabolic equations. In: Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991). Lecture Notes in Pure and Appl. Math. 155, New York:Dekker, pp.301–306, 1994 [30] Laurençot Ph.: Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions. Pacific J. Math. 230(2), 347–364 (2007) · Zbl 1221.35195 · doi:10.2140/pjm.2007.230.347 [31] Ladyzhenskaya O., Solonnikov V.A., Ural’ceva N.N.: Linear and Quasilinear Equations of Parabolic Type. Providence, RI, Amer. Math. Soc. (1967) [32] Laurençot Ph., Souplet Ph.: On the growth of mass for a viscous Hamilton-Jacobi equation. J. Anal. Math. 89, 367–383 (2003) · Zbl 1031.35054 · doi:10.1007/BF02893088 [33] Li Y.-X.: Stabilization towards the steady-state for a viscous Hamilton-Jacobi equation. Comm. Pure Appl. Anal. 8(6), 1917–1924 (2009) · Zbl 1179.35055 · doi:10.3934/cpaa.2009.8.1917 [34] Lieberman G.M.: The first initial-boundary value problem for quasilinear second order parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13(3), 347–387 (1986) [35] Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations. Research Notes in Mathematics, 69. Advanced Publishing Program. Boston, MA.-London:Pitman, 1982 [36] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications, 16. Basel, Birkhauser Verlag, 1995 [37] Matano H., Merle F.: On nonexistence of type II blowup for a supercritical nonlinear heat equation. Comm. Pure Appl. Math. 57(11), 1494–1541 (2004) · Zbl 1112.35098 · doi:10.1002/cpa.20044 [38] Merle F., Zaag H.: Stability of the blow-up profile for equations of the type u = ${\Delta }$u + |u| p-1 u. Duke Math. J. 86, 143–195 (1997) · Zbl 0872.35049 · doi:10.1215/S0012-7094-97-08605-1 [39] Quittner, P., Souplet, Ph.: Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basel Textbooks, Basel, Birkhäuser Verlag, 2007 [40] Souplet Ph.: Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions. Diff. Int. Eq. 15(2), 237–256 (2002) [41] Souplet Ph., Vázquez J.L.: Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Disc. Cont. Dyn. Syst. 14(1), 221–234 (2006) [42] Souplet Ph., Zhang Q.S.: Global solutions of inhomogeneous Hamilton-Jacobi equations. J. Anal. Math. 99, 355–396 (2006) · Zbl 1149.35050 · doi:10.1007/BF02789452 [43] Tersenov Al., Tersenov Ar.: Global solvability for a class of quasilinear parabolic problems. Indiana Univ. Math. J. 50, 1899–1913 (2001) · Zbl 1101.35331 · doi:10.1512/iumj.2001.50.2067 [44] Velázquez J.J.L.: Estimates on the (n 1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J. 42, 445–476 (1993) · Zbl 0802.35073 · doi:10.1512/iumj.1993.42.42021 [45] Weissler F.B.: Single point blow-up for a semilinear initial value problem. J. Diff. Eq. 55, 204–224 (1984) · Zbl 0555.35061 · doi:10.1016/0022-0396(84)90081-0 [46] Zhang Y.-C.: Singular dynamic interface equation from complex directed polymers. J. Phys. I France 2, 2175–2180 (1992) · doi:10.1051/jp1:1992274