Radial blow-up standing solutions for the semilinear wave equation
Maissâ Boughara
Universite Paris 13
Date(s) : 27/05/2025 iCal
10h00 - 11h00
We consider the following semilinear wave equation with subconformal
power nonlinearity in dimension $N$:
$$\partial^2_t U=\Delta U+|U|^{p-1}U,$$
where $p>1$ and if $N\geq 2$ then $p\leq 1+\frac{4}{N-1}$. We are able
construct a radial blow-up solution which converges, in similarity
variables, to a soliton near $(r_0, T (r_0))$ for a given $r_0>0$,
where $T(r_0)$ is the local blow-up time. For this purpose, we use a
modulation technique allowing us to kill the nonnegative modes of the
linearized operator of the equation around the soliton, in similarity
variables. We will also use some energy estimates from the one
dimensional case, with a new idea to control of some additional term we
have in our case. Combining all this with topological argument, we are
able to trap our error in some shrinking set for well chosen initial
data.
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