Generic behaviour of one-dimensional blow up patterns. (English) Zbl 0798.35081

Herrero, M. A.; Velázquez, J. J. L.

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 19, No. 3, 381-450 (1992).

This paper concerns the Cauchy problem \[ u_ t- u_{xx}= u^ p,\;x\in \mathbb{R},\;t>0,\quad u(x,0)= u_ 0(x),\;x\in \mathbb{R}, \] where \(p>0\) and \(u_ 0\) is continuous, nonnegative and compactly supported. The author shows that blow up consists generically in a single point \(\bar x\), and he gives an asymptotic expansion for \(\tau\to \infty\) of \(\Phi(y,\tau)\), where \[ \Phi(y,\tau):= (T- t)^{-(1/p-1)} u(x,t),\;y:= (x- \bar x)(T- t)^{-1/2},\;\tau:= -\log(T-t), \] and \(T\) is the blow up time.

Reviewer: L.Recke (Berlin)

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35K15	Initial value problems for second-order parabolic equations

Keywords:

generic behaviour; blow up

References:

[1]	A. Bressan , On the asymptotic shape of blow up , Indiana Univ. Math. J. , 39 , 4 ( 1990 ), 947 - 960 . MR 1087180 | Zbl 0705.35014 · Zbl 0705.35014 · doi:10.1512/iumj.1990.39.39045
[2]	A. Bressan , Stable blow up patterns , J. Differential Equations , 98 ( 1992 ), 57 - 75 . MR 1168971 | Zbl 0770.35010 · Zbl 0770.35010 · doi:10.1016/0022-0396(92)90104-U
[3]	J. Bebernes - D. Eberly , Mathematical problems from combustion theory , Appl. Math. Sci. 83 , Springer ( 1989 ). MR 1012946 | Zbl 0692.35001 · Zbl 0692.35001
[4]	X.Y. Chen - H. Matano , Convergence, asymptotic periodicity and finite point blow up in one-dimensional semilinear heat equations , J. Differential Equations , 78 ( 1989 ), 160 - 190 . MR 986159 | Zbl 0692.35013 · Zbl 0692.35013 · doi:10.1016/0022-0396(89)90081-8
[5]	J.W. Dold , Analysis of the early stage of thermal runaway , Quart. J. Mech. Appl. Math. 38 ( 1985 ), 361 - 387 . Zbl 0569.76079 · Zbl 0569.76079 · doi:10.1093/qjmam/38.3.361
[6]	A. Friedman , Partial Differential Equations of parabolic type , Robert E. Krieger Co ( 1983 ).
[7]	S. Filippas - R.V. Kohn , Refined asymptotics for the blow up of u t - \Delta u = up , Comm. Pure. Appl. Math. , XLV ( 1992 ), 821 - 869 . Zbl 0784.35010 · Zbl 0784.35010 · doi:10.1002/cpa.3160450703
[8]	A. Friedman - J.B. Mcleod , Blow up of positive solutions of semilinear heat equations , Indiana Univ. Math. J. , 34 ( 1985 ), 425 - 447 . MR 783924 | Zbl 0576.35068 · Zbl 0576.35068 · doi:10.1512/iumj.1985.34.34025
[9]	H. Fujita , On the blowing up of solutions of the Cauchy problem for ut = \Delta u+u1+\alpha , J. Fac. Sci. Univ. Tokyo Sect. I , 13 ( 1966 ), 109 - 124 . Zbl 0163.34002 · Zbl 0163.34002
[10]	V.A. Galaktionov - M.A. Herrero - J.J.L. Velázquez , The space structure near a blow up point for semilinear heat equations , Soviet J. Comput. Math. and Math. Physics , 31 ( 1991 ), 399 - 411 . MR 1107061 | Zbl 0747.35014 · Zbl 0747.35014
[11]	Y. Giga - R.V. Kohn , Asymptotically self-similar blow up of semilinear heat equations , Comm. Pure Appl. Math. , 38 ( 1985 ), 297 - 319 . MR 784476 | Zbl 0585.35051 · Zbl 0585.35051 · doi:10.1002/cpa.3160380304
[12]	Y. Giga - R.V. Kohn , Characterizing blow up using similarity variables , Indiana Univ. Math. J. , 36 ( 1987 ), 1 - 40 . MR 876989 | Zbl 0601.35052 · Zbl 0601.35052 · doi:10.1512/iumj.1987.36.36001
[13]	Y. Giga - R.V. Kohn , Nondegeneracy of blow up for semilinear heat equations , Comm. Pure Appl. Math. , 42 ( 1989 ), 845 - 884 . MR 1003437 | Zbl 0703.35020 · Zbl 0703.35020 · doi:10.1002/cpa.3160420607
[14]	V.A. Galaktionov - S.A. Posashkov , Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations , Differential Equations , 22 , 7 ( 1986 ), 116 - 1183 . MR 853803 | Zbl 0632.35028 · Zbl 0632.35028
[15]	M.A. Herrero - J.J.L. Velázquez , Blow up behaviour of one-dimensional semilinear parabolic equations , Ann. Inst. H. Poincaré , to appear. Numdam | MR 1220032 | Zbl 0813.35007 · Zbl 0813.35007
[16]	M.A. Herrero - J.J.L. Velázquez , Flat blow up in one-dimensional semilinear heat equations , Differential Integral Equations , Vol. 5 , n^\circ 5 ( 1992 ), 973 - 997 . MR 1171974 | Zbl 0767.35036 · Zbl 0767.35036
[17]	M.A. Herrero - J.J.L. Velázquez , Blow up profiles in one-dimensional, semilinear parabolic problems , Comm. Partial Differential Equations , 17 , 182 , ( 1992 ), 205 - 219 . MR 1151261 | Zbl 0772.35027 · Zbl 0772.35027 · doi:10.1080/03605309208820839
[18]	M.A. Herrero - J.J.L. Velázquez , Plane structures in thermal runaway , Israel J. Math. , to appear. MR 1231197 | Zbl 0813.35035 · Zbl 0813.35035 · doi:10.1007/BF02764836
[19]	L.M. Hocking - K. Stewartson - J.T. Stuart , A nonlinear instability burst in plane parallel flow , J. Fluid Mech. J 1 ( 1972 ), 705 - 735 . Zbl 0232.76053 · Zbl 0232.76053 · doi:10.1017/S0022112072001326
[20]	J.L. Lions - B. Malgrange , Sur l’unicité retrograde , Math. Scand. 8 ( 1960 ), 277 - 286 . MR 140855 | Zbl 0126.12202 · Zbl 0126.12202
[21]	F. Merle , Solution of a nonlinear heat equation with arbitrarily given blow up points , Comm. Pure. Appl. Math. , to appear. MR 1151268 | Zbl 0785.35012 · Zbl 0785.35012 · doi:10.1002/cpa.3160450303
[22]	C.E. Müller - F.B. Weissler , Single point blow up for a general semilinear heat equation , Indiana Univ. Math. J. , 34 ( 1983 ), 881 - 913 . MR 808833 | Zbl 0597.35057 · Zbl 0597.35057 · doi:10.1512/iumj.1985.34.34049
[23]	J.J.L. Velázquez , Local behaviour near blow up points for semilinear parabolic equations , J. Differential Equations , to appear. MR 1251859 | Zbl 0798.35023 · Zbl 0798.35023 · doi:10.1006/jdeq.1993.1113
[24]	F.B. Weissler , Single point blow up of semilinear initial value problems , J. Differential Equations , 55 ( 1984 ), 204 - 224 . MR 764124 | Zbl 0555.35061 · Zbl 0555.35061 · doi:10.1016/0022-0396(84)90081-0