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Self-similar solutions and semi-linear wave equations in Besov spaces. (English) Zbl 0979.35106
The Cauchy problem for the semi-linear wave equation $$\square u =\pm u^p$$, $$u(x,0) = u_0(x)$$, $$\partial_tu(x,0) = u_1(x)$$ on $$\mathbb{R}^n$$, $$n\geq 2$$, is regarded. The existence and regularity theorems for the global and semi-similar solutions are proved.
Let $$s_p =\frac n2-\frac{2}{p-1}$$, $$p^*=\frac{n+3}{n-1}$$ and $$\dot H^{s_p}_p (\mathbb{R}^n)$$ $$(B^{s,q}_p(\mathbb{R}^n))$$ be a homogeneous Sobolev (Besov) space, respectively. It is proved that if $$p > p^*$$ and $$u_i\in \dot B^{s_p-i,\infty}_2(\mathbb{R}^n)$$, $$i = 0,1$$, then there exists a global solution of the above Cauchy problem. Moreover, the regularity properties of the solution are formulated using the function spaces of the type related to the homogeneous Besov spaces. The sufficient conditions for the uniqueness of the solution are given. If one has a better regularity of the initial data, $$u_i\in\dot H^{s_p-i}_2(\mathbb{R}^n)$$, $$i = 0,1$$, then the better regularity of solution can be proved. For $$u_i\in\dot B^{s_p-1}_2 (\mathbb{R}^n)$$, $$i = 0,1$$, $$1\leq q <\infty$$, the local in time theory is formulated.
If the initial data are homogeneous $$u_i = \phi_i(x/x)|x|^{\frac{2}{p-1}+i}$$, $$\phi_i \in \dot H^{s_p-i}_2(S^{n-1})$$, $$i = 0,1$$, then there exists a unique self-similar solution $$u(x,t) = \frac 1t U(\frac xt)$$ and the smoothness of $$U$$ can be described in terms of the homogeneous Besov spaces.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations
##### Keywords:
sufficient conditions for the uniqueness
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