Finite time blow up for critical wave equations in high dimensions

By Borislav Yordanov, Qi S. Zhang. 2004

252 citations recorded by the source corpus.

We prove that solutions to the critical wave equation (1.1) with dimension n⩾4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form Δu-∂t2u+|u|p=0. The rest of the cases, the lower-dimensional case n⩽3, and the sub or super critical cases were settled many years earlier by the work of several authors.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e53dbd59cac62310150aae61dec8a84cb300cdcb

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Citations in the corpus

A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass (2019, methodology): “k 0 = k 0(k 1) >0 is suﬃciently large. Then, if (2.3) holds with a= q−2 p−1 for some T 0 ∈ [0,T),then T<∞. For the proof of the previous result one can see [27, 35] for the subcritical case and [39, 6] for the critical case. The strategy in the proof of Theorem 1.1 is to apply Kato’s lemma to ”
Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case (2019, methodology): “Then, in Section 4 we apply an iteration method associated with the test function introduced in [45] to prove the blow – up of energy solutions in the subcritical case.”
A shift in the Strauss exponent for semilinear wave equations with a not effective damping (2015, background): “ global (in time) small data solutions is p0(n). More precisely, if 1 <p≤ p0(n), then solutions to (2) blow-up in ﬁnite time for a suitable choice of initial data (see [9], [13], [14], [20], [21], [24]), whereas for p∈ (p0(n),(n+ 3)/(n− 1)] a unique global (in time) small data solution exists (se”
Life-Span of Solutions to Critical Semilinear Wave Equations (2011, methodology): “When n ≥ 4, the blow-up problem for the critical wave equations problem was solved by Yordanov and Zhang [21] and Zhou [25] by different methods respectively.”
Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation (2017, methodology): “More precisely, the following result, which proof can be found in the paper [27], holds for any smooth solution to wt t − w = G(t, x, w) with nonnegative G.”
Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions* (2007, result): “After the completion of this paper, we received a preprint of [10], similar result is proved by different method.”
On the global solution problem for semilinear generalized Tricomi equations, I (2015, background): “ve, blowup for the solutionuof (1.5) has been shown, while, for p>p1(n) and some other restrictions on the power p, global existence of small data solution uof (1.5) has also been established (see [5, 6, 7, 11, 13, 23, 24, 32, 33] and the references therein). Remark 1.3. For the cases of n= 1,2 o”
Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data (2017, background): “In particular, the threshold exponent for dividing blowup phenomena and global existence of small solutions is clarified as p0(N) (for example, see Yordanov–Zhang [15] and Zhou [16]), where p0(n) is called Strauss exponent and is given as the positive root of the quadratic equation (n − 1)p2 − (n + ”