## Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data.(English)Zbl 0803.35068

From the introduction: We consider the Cauchy problem for semilinear heat equations on $$\mathbb{R}^ n$$ of the following form: $(1) \quad {\partial u \over \partial t} (t,x) = \Delta u(t,x) + f \bigl( u(t,x) \bigr) \quad \text{in} \quad ]0,\infty [\times \mathbb{R}^ n, \qquad (2) \quad u(0,x) = a(x) \quad \text{on} \quad \mathbb{R}^ n,$ where $$f(\sigma)$$ is a locally Lipschitz continuous function on $$\mathbb{C}$$ satisfying (3) $$\bigl | f (\sigma) - f (\rho) \bigr | \leq C | \sigma - \rho | \bigl( 1+ | \sigma |^{\gamma - 1} + | \rho |^{\gamma - 1} \bigr)$$, or, more strongly, (4) $$f(0) = 0$$ and $$| f (\sigma) - f(\rho) | \leq C | \sigma - \rho | (| \sigma |^{\gamma - 1} + | \rho |^{\gamma - 1})$$ for every $$\sigma$$, $$\rho \in \mathbb{C}$$ with some constants $$\gamma>1$$ and $$C>0$$.
We also consider the Cauchy problem for the Navier-Stokes equation on $$\mathbb{R}^ n$$ for $$n \geq 2$$ of the following form: ${\partial u \over \partial t} - \Delta_ xu + (u \cdot \nabla_ x) u + \nabla_ x \pi = 0 \quad \text{in} \quad ]0, \infty [\times \mathbb{R}^ n, \tag{5}$
$(6) \qquad \nabla_ x \cdot u = 0 \quad \text{in} \quad ]0, \infty [\times \mathbb{R}^ n, \qquad (7) \qquad u(0,x) = a(x) \quad \text{on} \quad \mathbb{R}^ n.$ The purpose of this paper is to construct new function spaces in the same way as the Besov spaces, based on the Morrey spaces in place of the standard $$L^ p$$-spaces, and to show that, if the initial data $$a(x)$$ belongs to some function spaces above and its norm is sufficiently small, then the Cauchy problems (1)–(2) and (5)–(7) admit unique time-global strong solutions with a bound near $$t=0$$, provided that the function $$f(\sigma)$$ in (1) satisfies (4) with some constant $$\gamma > 1 + 2/n$$. We also introduce a local version of the above function spaces, and show that the above Cauchy problems, with $$f(\sigma)$$ satisfying (3) in (1), admit unique time-local strong solutions with a bound near $$t=0$$ for initial data in the local function spaces under an additional assumption.

### MSC:

 35K55 Nonlinear parabolic equations 35Q30 Navier-Stokes equations 35K15 Initial value problems for second-order parabolic equations 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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### References:

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