The Cauchy problem for critical and subcritical semilinear parabolic equations in \(L^r\). I (Q5960867)
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scientific article; zbMATH DE number 1730541
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The Cauchy problem for critical and subcritical semilinear parabolic equations in \(L^r\). I |
scientific article; zbMATH DE number 1730541 |
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The Cauchy problem for critical and subcritical semilinear parabolic equations in \(L^r\). I (English)
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17 October 2002
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global existence
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time-space \(L^p\)-\(L^r\) estimates
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The paper deals with the semilinear Cauchy problem NEWLINE\[NEWLINE u_t+(1+ib)(-\Delta)^{\theta/2}u=\sum_{0\leq |\alpha_i|<\theta} c(\alpha_i) D^{\alpha_i} f_{\alpha_i}(u),\qquad u(0,x)=\varphi(x),NEWLINE\]NEWLINE where \(u(t,x)\) is a complex-valued function defined on \([0,\infty)\times{\mathbb{R}}^n,\) \(b\in{\mathbb{R}},\) \(0<\theta<\infty,\) and \(c(\alpha_i)\)'s are complex constants. Global existence results are given for the above problem when \(\varphi\in L^r\) with small norm.
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