×

Single point blow-up for a semilinear initial value problem. (English) Zbl 0555.35061

Since the pioneering work of H. Fujita [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.340)] the non-existence of global solutions to semilinear parabolic equations raised considerable interest [among the main results in this direction, recall J. M. Ball, Q. J. Math., Oxford II. Ser. 28, 473-486 (1977; Zbl 0377.35037), and of the author, Isr. J. Math. 38, 29-40 (1981; Zbl 0476.35043)]. In the present, stimulating paper, the author addresses the equation \[ (*)\quad u_ t(t,x)=u_{xx}(t,x)+u^{\gamma}(t,x)\quad in\quad (-R,R)\times {\mathbb{R}}^+,\quad u(r,-R)=u(t,R)=0,\quad t>0,\quad u(0,x)=\phi (x). \] It is well-known that for every \(\phi \in C_ 0([-R,R])\) there is a unique solution to (*) defined in a maximal time interval [0,T). It is also known that if \(\phi\) is positive and ”large” enough, then \(T<\infty\), moreover, as the author showed in the paper quoted above, if \(p\geq 1\) and \(p>(\gamma -1)/2,\) then \(\| u(t)\|_ p\to \infty\) as \(t\to T.\)
Here the author considers initial data of the form \(\phi =k\psi\), where \(\psi\) is a positive solution of the associated stationary problem, and \(k>1\) is chosen so large that the associated existence time is finite. He then proves that if \(\gamma >2\) and is ”large”, then, as t approaches T, both u(t,x) and \(u_ x(t,x)\) have a finite limit for all \(x=0:\) in other words, blow-up occurs only at the point \(x=0\). The proof requires a number of steps, carried out in detail in a terse and comprehensive way. Since a similar single point blow-up behaviour prevails, too, for the purely reactive problem \(u_ t(t,x)=u^{\gamma}(t,x)\) in \((-R,R)\times {\mathbb{R}}^+,\) the author asks whether other properties of the latter problem are shared by (*), such as the boundedness of the p-norm of the solution as \(t\to T\) for \(1\leq p\leq (\gamma -1)/2,\) and the approximate representation of the solution \(u(T,x)\sim C| x|^{-2(\gamma - 1)}\) for X near 0. These items represent so far open, challenging questions. (Another open question is how far the single point blow-up depends on the fact that the positive initial function has itself a maximum at zero: had it two maxima, at, say, \(\pm R/r\) \((r>1)\), would single point blow-up still occurr ?)
Reviewer: P.de Mottoni

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ball, J. M., Remarks on blow-up and nonexistence theorems for nonlinear evoluation equations, Quart. J. Math. Oxford Ser., 28, 473-486 (1977) · Zbl 0377.35037
[2] Ball, J. M., Finite time blow-up in nonlinear problems, (Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin. Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin, Madison, October 1977 (1978), Academic Press: Academic Press New York) · Zbl 0472.35060
[3] Cazenave, T., Equations de Schrödinger non linéaires en dimension deux, (Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979)), 327-346 · Zbl 0428.35021
[4] Glassey, R., Blow-up theorems for nonlinear wave equations, Math. Z., 132, 183-203 (1973) · Zbl 0247.35083
[5] Glassey, R., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, (J. Math. Phys., 18 (1977)), 1794-1797 · Zbl 0372.35009
[6] R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, preprint.; R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, preprint. · Zbl 0438.35045
[7] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = Δu + u^{1 + α} \), J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 13, 109-124 (1966) · Zbl 0163.34002
[8] Fujita, H., On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, (Proc. Sympos. Pure Math., 18 (1968)), 138-161, (Part I) · Zbl 0228.35048
[9] John, F., Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28, 235-268 (1979) · Zbl 0406.35042
[10] Levine, H. A., Some nonexistence and instability theorems of formally parabolic equations of the form \(Pu^t=−Au+F\)(u), Arch. Rational Mech. Anal., 51, 371-386 (1973) · Zbl 0278.35052
[11] Levine, H. A.; Payne, L. E., On the nonexistence of global solutions to some abstract Cauchy problems of standard and non standard types, Rend. Mat. (2), 8, 413-428 (1975) · Zbl 0311.34074
[12] H. A. Levine and P. E. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, preprint.; H. A. Levine and P. E. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, preprint. · Zbl 0487.34003
[13] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0153.13602
[14] Segal, I., Non-linear semi-groups, Ann. of Math., 78, 339-364 (1963) · Zbl 0204.16004
[15] T. C. Sideris, Nonexistence of global solutions to semi-linear wave equations in high dimensions, J. Differential Equations52, 378-406.; T. C. Sideris, Nonexistence of global solutions to semi-linear wave equations in high dimensions, J. Differential Equations52, 378-406. · Zbl 0555.35091
[16] T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, preprint.; T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, preprint. · Zbl 0534.35069
[17] Strauss, W. A., The nonlinear Schrödinger equation, (Proceedings, Conf. Cont. Mech. and PDE (1978), North-Holland: North-Holland Amsterdam), 452-465
[18] Strauss, W. A., Everywhere defined wave operators, (Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin. Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin, Madison, October 1977 (1978), Academic Press: Academic Press New York) · Zbl 0466.47005
[19] Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034
[20] Weissler, F. B., Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38, 29-40 (1981) · Zbl 0476.35043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.