^{1,a)}

### Abstract

The aim of this paper is threefold. The first and also main purpose is to provide numerical evidence for the conjecture proposed by Bizoń *et al.* [“Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation,” J. Math. Phys.52, 103703 (2011)]10.1063/1.3645363 that the blowup evolution of spherically symmetric semilinear Klein-Gordon equations is similar to the evolution of spherically symmetric semilinear wave equations, i.e., the mass term can be neglected when the amplitude of a solution grows. The second aim is to describe the relationship between different types of blowup for energy critical semilinear wave equations. The third goal is to present numerical evidence for the fact that the special class of self-similar profiles of semilinear wave equations found by Kycia [“On self-similar solutions of semilinear wave equations in higher space dimensions,” Appl. Math Comput.217, 9451–9466 (2011)]10.1016/j.amc.2011.04.039 play the same role in the evolution of semilinear wave and Klein-Gordon equations as the previously known ordinary profiles. All the results are presented in spherical symmetry.

The author thanks Piotr Bizoń and Tadeusz Chmaj for suggestions and support during the work, not only on this paper, and Edward Malec and Patryk Mach for suggestions on the Lane-Emden equation. The author is also grateful to the organizers of the conference, “On formal and analytic solutions of differential and difference equations” in Bedlewo, Poland in 2011 for the opportunity to participate in this event. The research was carried out with the ,“Deszno” supercomputer purchased, thanks to the financial support of the european regional development fund in the framework of the Polish innovation economy operational program (Contract No. POIG.02.01.00-12-023/08). The work on this paper was partially supported by the Polish Ministry of Science and Higher Education (Grant No. NN202 079235) and by the National Science Centre (Grant No. NN202 030740).

I. INTRODUCTION

II. STABILITY ANALYSIS

A. Self-similar solutions

B. Static solutions—critical case

III. GENERIC BLOWUP

IV. NEAR-THRESHOLD BEHAVIOR

A. Critical case

B. Supercritical case

V. CONCLUSIONS

### Key Topics

- Attractors
- 23.0
- Partial differential equations
- 10.0
- Solution processes
- 10.0
- Eigenvalues
- 8.0
- Wave equations
- 8.0

## Figures

Blowup out of the origin for NLW *n* = 3, *p* = 7. Upper panel: Normalized spatial profiles . Bottom panel: Near-blowup asymptotics is approximated by *U* _{0} profile. From fit *U*(*t*, *r* _{ max })^{−3} ≈ *a*(*T* − *t*) we get the blowup time *T* = 0.0404829 and *a* = 1.49855, which approximately equals .

Blowup out of the origin for NLW *n* = 3, *p* = 7. Upper panel: Normalized spatial profiles . Bottom panel: Near-blowup asymptotics is approximated by *U* _{0} profile. From fit *U*(*t*, *r* _{ max })^{−3} ≈ *a*(*T* − *t*) we get the blowup time *T* = 0.0404829 and *a* = 1.49855, which approximately equals .

Blowup out of the origin for NLKG *n* = 3, *p* = 7. Upper panel: Normalized spatial profiles . Bottom panel: Near-blowup asymptotics is approximated by *U* _{0} profile of NLW providing that Conjecture 1 is true. From fit *U*(*t*, *r* _{ max })^{−3} ≈ *a*(*T* − *t*) we get the blowup time *T* = 0.0405039 and *a* = 1.49994 which approximately equals .

Blowup out of the origin for NLKG *n* = 3, *p* = 7. Upper panel: Normalized spatial profiles . Bottom panel: Near-blowup asymptotics is approximated by *U* _{0} profile of NLW providing that Conjecture 1 is true. From fit *U*(*t*, *r* _{ max })^{−3} ≈ *a*(*T* − *t*) we get the blowup time *T* = 0.0405039 and *a* = 1.49994 which approximately equals .

Central blowup for NLW *n* = 3, *p* = 7. Upper panel: Near-blowup asymptotics is approximated by . From the fit *c* _{1} = 1.20094, *T* = 3.18068. Bottom panel: Spatial profile near the blowup with the same *T*, *c* _{1} and δ = 0.001154.

Central blowup for NLW *n* = 3, *p* = 7. Upper panel: Near-blowup asymptotics is approximated by . From the fit *c* _{1} = 1.20094, *T* = 3.18068. Bottom panel: Spatial profile near the blowup with the same *T*, *c* _{1} and δ = 0.001154.

Central blowup for NLKG *n* = 3, *p* = 7. Upper panel: Near-blowup asymptotics, as Conjecture 1 suggest, is approximated by . From the fit *c* _{1} = 1.51724, *T* = 3.15336. Bottom panel: Spatial profile near the blowup with the same *T*, *c* _{1} and δ = 0.00337.

Central blowup for NLKG *n* = 3, *p* = 7. Upper panel: Near-blowup asymptotics, as Conjecture 1 suggest, is approximated by . From the fit *c* _{1} = 1.51724, *T* = 3.15336. Bottom panel: Spatial profile near the blowup with the same *T*, *c* _{1} and δ = 0.00337.

Upper panel: Sketch of initial parameters plane for critical NLW *n* = 3, *p* = 5; *s* = 2. Edge 2 corresponds to the type II blowup and edge 4 to the static solution. Area 3 presents dispersion and areas 1 and 5 generic blowup. Near edge 4 there is a dispersion or generic/ODE blowup after a departure from the static intermediate attractor and near edge 2 there is a dispersion or generic/ODE blowup after departure from the type II blowup intermediate attractor. Scanning with 10^{−6} resolution does not reveal any exotic structures of the edges, e.g., fractals, etc. The cusp of 4 is located around the point (*x* _{0}; *A*) ≈ (0.5634; 0.2025). Bottom panel: The left figure presents *U*(*t*, *r* = 0) near edge 4. There is plateau that corresponds to the vicinity of the static solution. The right figure presents the solutions at *r* = 0 near edge 2 of the type II blowup. There is no static episode in the evolution, and instead, both solutions before the departure approach the type II blowup solution.

Upper panel: Sketch of initial parameters plane for critical NLW *n* = 3, *p* = 5; *s* = 2. Edge 2 corresponds to the type II blowup and edge 4 to the static solution. Area 3 presents dispersion and areas 1 and 5 generic blowup. Near edge 4 there is a dispersion or generic/ODE blowup after a departure from the static intermediate attractor and near edge 2 there is a dispersion or generic/ODE blowup after departure from the type II blowup intermediate attractor. Scanning with 10^{−6} resolution does not reveal any exotic structures of the edges, e.g., fractals, etc. The cusp of 4 is located around the point (*x* _{0}; *A*) ≈ (0.5634; 0.2025). Bottom panel: The left figure presents *U*(*t*, *r* = 0) near edge 4. There is plateau that corresponds to the vicinity of the static solution. The right figure presents the solutions at *r* = 0 near edge 2 of the type II blowup. There is no static episode in the evolution, and instead, both solutions before the departure approach the type II blowup solution.

Upper panel: Convergence of the solution of critical NLW to a normalized static one for the type II blowup for *n* = 3, *p* = 5 in the intermediate region. Snapshots of the profiles for a given time are scaled , where λ(*t*) = 1/*U*(*t*, *r* = 0)^{2}. Bottom panel: The departure from the type II intermediate attractor. One can note that there exist two times of a blowup. The type II intermediate attractor has the blowup time *T* _{1}, but before that time the unstable mode takes over, and the solution blows up in a generic/ODE way with the blowup time *T* _{2} < *T* _{1}.

Upper panel: Convergence of the solution of critical NLW to a normalized static one for the type II blowup for *n* = 3, *p* = 5 in the intermediate region. Snapshots of the profiles for a given time are scaled , where λ(*t*) = 1/*U*(*t*, *r* = 0)^{2}. Bottom panel: The departure from the type II intermediate attractor. One can note that there exist two times of a blowup. The type II intermediate attractor has the blowup time *T* _{1}, but before that time the unstable mode takes over, and the solution blows up in a generic/ODE way with the blowup time *T* _{2} < *T* _{1}.

Upper panel: Convergence of the solution of critical NLKG to normalized wave static one for blowup for *n* = 3, *p* = 5 in the intermediate region. Snapshots of the profiles for a given time are scaled , where λ(*t*) = 1/*U*(*t*, *r* = 0)^{2}. Bottom panel: Corresponding values of *U*(*t*, *r* = 0)^{−2}.

Upper panel: Convergence of the solution of critical NLKG to normalized wave static one for blowup for *n* = 3, *p* = 5 in the intermediate region. Snapshots of the profiles for a given time are scaled , where λ(*t*) = 1/*U*(*t*, *r* = 0)^{2}. Bottom panel: Corresponding values of *U*(*t*, *r* = 0)^{−2}.

Upper panel: Convergence of the solution of supercritical NLW to *u* _{1} profile in the transient phase for *n* = 3, *p* = 7. From theory it is known that . Bottom panel: The departure from the intermediate attractor *u* _{1}. Near intermediate attractor .

Upper panel: Convergence of the solution of supercritical NLW to *u* _{1} profile in the transient phase for *n* = 3, *p* = 7. From theory it is known that . Bottom panel: The departure from the intermediate attractor *u* _{1}. Near intermediate attractor .

Upper panel: Convergence of the NLKG solution to the NLW *u* _{1} profile in the transient phase for *n* = 3, *p* = 7. From Conjecture 1, it is supposed that as for NLW. Bottom panel: The departure from the NLW intermediate attractor *u* _{1}. Near intermediate attractor .

Upper panel: Convergence of the NLKG solution to the NLW *u* _{1} profile in the transient phase for *n* = 3, *p* = 7. From Conjecture 1, it is supposed that as for NLW. Bottom panel: The departure from the NLW intermediate attractor *u* _{1}. Near intermediate attractor .

Upper panel: Convergence of the solution of supercritical NLW to *u* _{1} profile in the transient phase for *n* = 4, *p* = 5. From theory it is known, that . Bottom panel: In the intermediate region (*T* − *t*)^{1/2} *U*(*t*, *r*/(*T* − *t*)) ≈ *u* _{1}(ρ). One can see that the intermediate attractor of NLW matches *u* _{1} also at ρ = 1.

Upper panel: Convergence of the solution of supercritical NLW to *u* _{1} profile in the transient phase for *n* = 4, *p* = 5. From theory it is known, that . Bottom panel: In the intermediate region (*T* − *t*)^{1/2} *U*(*t*, *r*/(*T* − *t*)) ≈ *u* _{1}(ρ). One can see that the intermediate attractor of NLW matches *u* _{1} also at ρ = 1.

Upper panel: Convergence of the NLKG solution to the NLW *u* _{1} profile in the transient phase for *n* = 4, *p* = 5. From Conjecture 1, it is supposed that as for NLW. Bottom panel: As for NLW in the intermediate region (*T* − *t*)^{1/2} *U*(*t*, *r*/(*T* − *t*)) ≈ *u* _{1}(ρ). As previously the intermediate attractor of NLKG matches *u* _{1} also at ρ = 1.

Upper panel: Convergence of the NLKG solution to the NLW *u* _{1} profile in the transient phase for *n* = 4, *p* = 5. From Conjecture 1, it is supposed that as for NLW. Bottom panel: As for NLW in the intermediate region (*T* − *t*)^{1/2} *U*(*t*, *r*/(*T* − *t*)) ≈ *u* _{1}(ρ). As previously the intermediate attractor of NLKG matches *u* _{1} also at ρ = 1.

## Tables

Eigenvalues for a few unstable modes of *u* _{1} for supercritical NLW. The values of *u* _{1}(ρ = 0) are also presented. They were obtained using the numerical methods described in Ref. 18. One can observe that the eigenvalues decrease when *n* and *p* increase. Therefore, for higher *n* and *p* the unstable solutions “lives” longer, because the unstable mode grows more slowly. In this paper, we restricted ourselves to odd *p*. But we think it is appropriate to indicate the same structure for supercritical even *p*, as well as for critical even *p* in Table II.

Eigenvalues for a few unstable modes of *u* _{1} for supercritical NLW. The values of *u* _{1}(ρ = 0) are also presented. They were obtained using the numerical methods described in Ref. 18. One can observe that the eigenvalues decrease when *n* and *p* increase. Therefore, for higher *n* and *p* the unstable solutions “lives” longer, because the unstable mode grows more slowly. In this paper, we restricted ourselves to odd *p*. But we think it is appropriate to indicate the same structure for supercritical even *p*, as well as for critical even *p* in Table II.

Eigenvalues for unstable modes *v* _{1} for critical NLW. Here . Similarly as for subcritical cases, the eigenvalues are smaller for larger *n* values.

Eigenvalues for unstable modes *v* _{1} for critical NLW. Here . Similarly as for subcritical cases, the eigenvalues are smaller for larger *n* values.

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