The role of critical exponents in blowup theorems.

*(English)*Zbl 0706.35008The survey presents some basic results on the critical exponents for nonlinear evolution problems. One typical example is the following nonlinear problem for the heat equation
\[
(F)\quad u_ t=\Delta u+u^ p,\quad x\in {\mathbb{R}}^ N,\quad t>0,\quad u(0,x)=u_ 0(x),\quad x\in {\mathbb{R}}^ N,
\]
where \(\Delta\) denotes the N-dimensional Laplace operator. A result due to Fujita guarantees that for the critical exponent \(p_ c(N)=1+2/N\) the following two statements are fulfilled.

(A) If \(1<p<p_ c(N)\), then the only nonnegative global (in time) solution of (F) is \(u=0.\)

(B) If \(p>p_ c(N)\), then there exists a global positive solution of (F), if the initial data are sufficiently small.

The survey is divided into four sections. The first section deals with some extensions of the problem (F). The first part 1.1 of this section is devoted to the cases of other geometries, various linear dissipative terms or other reaction terms. More precisely, if D \((\subset {\mathbb{R}}^ N)\) is any bounded or unbounded domain, then in the place of (F) the author considers the initial boundary value problem \[ (D)\quad u_ t=\Delta u+u^ p,\quad (x,t)\in D\times (0,T),\quad u(0,x)=u_ 0(x),\quad x\in D,\quad u(t,x)=0,\quad (x,t)\in \partial D\times (0,T), \] or the following generalization of (D) \((GD)\quad u_ t=\sum^{N}_{i,j=1}(a_{ij}(t,x)u_{x_ i})_{x_ j}+\sum^{N}_{i=1}b_ i(t,x)u_{x_ i}+u^ p\) (p\(\leq 1)\), \(u(0,x)=u_ 0(x)\), \(x\in D\), \(u(t,x)=0\), (x,t)\(\in \partial D\times (0,T)\), where the coefficients of the linear operator of the right-hand side are uniformly bounded in \(D\times (0,\infty)\). A result due to Meier asserts that a critical exponent \(p_ c(GD)\geq 1\) exists. An explicit representation of \(p_ c(GD)\) or \(p_ c(D)\) is known for special cases of D. For example, if D is the “orthant” \(D_ k=\{x\in {\mathbb{R}}^ N\); \(x_ 1>0,...\), \(x_ k>0\}\), then we have \(p_ c(D_ k)=1+2/(k+N)\) according to a result due to Meier. Another case studied in the first part of section 1 is the case of a cone D with a vertex at the origin. An explicit representation of \(p_ c(D)\) is found by Levine, Bandle and Meier.

Another problem close to (F) is the Dirichlet problem for the nonlinear heat equation in which \(u^ p\) is replaced by \(| u|^{p-1}u\). In this case one is interested in real valued solutions. This part contains also a summary on the results for \(u_ t=\Delta u+| x|^{\sigma}u^ p\) or \(u_ t=\Delta u+t^ k| x|^{\sigma}u^ p\) and the dependence of the critical exponent on k, \(\sigma\), p.

For the general case of the problem (GD), where D has bounded complement upper and lower bounds for \(p_ c(GD)\) are found according to the results of Bandle and Levine.

Part 1.2 of section 1 summarizes the results on the problem \(u_ t=A(u)+u^ p\), where A(u) is in general a nonlinear dissipative term. Various special choices of A(u) are studied. For example a typical choice of A is given by \[ A(u)=div\{\frac{\nabla_ xu}{(1+| \nabla_ xu|^ 2)^{1/2}}\} \] representing the mean curvature operator. Another choice of A was considered by Galaktionov, \(A(u)=\sum^{N}_{i=1}\partial_{x_ i}(| \nabla u|^{\sigma} \partial_{x_ i}u).\)

Part 1.3 of section 1 contains a summary of results for bounded domains D, while the part 1.4 is devoted to systems of equations. For example \(u_ t=\Delta u+v^ p\), \(v_ t=\Delta v+u^ p.\)

In section II the author considers the nonlinear Schrödinger equation \[ (NLS)\quad iu_ t+\Delta u+| u|^{p-1}u=0,\quad x\in {\mathbb{R}}^ N,\quad t>0,\quad u(0,x)=u_ 0(x). \] For this problem the critical exponent is \(p_{nls}(N)=1+4/N.\)

In section III the nonlinear wave equation \(u_{tt}=\Delta u+| u|^ p\), \(u(0,x)=u_ 0(x)\), \(u_ t(0,x)=u_ 1(x)\) as well as the critical exponent for this equation are examined. The critical exponent is the larger root of the quadratic equation \((N-1)p^ 2-(N+1)p-2=0.\)

Finally, in section IV some concluding remarks are discussed.

(A) If \(1<p<p_ c(N)\), then the only nonnegative global (in time) solution of (F) is \(u=0.\)

(B) If \(p>p_ c(N)\), then there exists a global positive solution of (F), if the initial data are sufficiently small.

The survey is divided into four sections. The first section deals with some extensions of the problem (F). The first part 1.1 of this section is devoted to the cases of other geometries, various linear dissipative terms or other reaction terms. More precisely, if D \((\subset {\mathbb{R}}^ N)\) is any bounded or unbounded domain, then in the place of (F) the author considers the initial boundary value problem \[ (D)\quad u_ t=\Delta u+u^ p,\quad (x,t)\in D\times (0,T),\quad u(0,x)=u_ 0(x),\quad x\in D,\quad u(t,x)=0,\quad (x,t)\in \partial D\times (0,T), \] or the following generalization of (D) \((GD)\quad u_ t=\sum^{N}_{i,j=1}(a_{ij}(t,x)u_{x_ i})_{x_ j}+\sum^{N}_{i=1}b_ i(t,x)u_{x_ i}+u^ p\) (p\(\leq 1)\), \(u(0,x)=u_ 0(x)\), \(x\in D\), \(u(t,x)=0\), (x,t)\(\in \partial D\times (0,T)\), where the coefficients of the linear operator of the right-hand side are uniformly bounded in \(D\times (0,\infty)\). A result due to Meier asserts that a critical exponent \(p_ c(GD)\geq 1\) exists. An explicit representation of \(p_ c(GD)\) or \(p_ c(D)\) is known for special cases of D. For example, if D is the “orthant” \(D_ k=\{x\in {\mathbb{R}}^ N\); \(x_ 1>0,...\), \(x_ k>0\}\), then we have \(p_ c(D_ k)=1+2/(k+N)\) according to a result due to Meier. Another case studied in the first part of section 1 is the case of a cone D with a vertex at the origin. An explicit representation of \(p_ c(D)\) is found by Levine, Bandle and Meier.

Another problem close to (F) is the Dirichlet problem for the nonlinear heat equation in which \(u^ p\) is replaced by \(| u|^{p-1}u\). In this case one is interested in real valued solutions. This part contains also a summary on the results for \(u_ t=\Delta u+| x|^{\sigma}u^ p\) or \(u_ t=\Delta u+t^ k| x|^{\sigma}u^ p\) and the dependence of the critical exponent on k, \(\sigma\), p.

For the general case of the problem (GD), where D has bounded complement upper and lower bounds for \(p_ c(GD)\) are found according to the results of Bandle and Levine.

Part 1.2 of section 1 summarizes the results on the problem \(u_ t=A(u)+u^ p\), where A(u) is in general a nonlinear dissipative term. Various special choices of A(u) are studied. For example a typical choice of A is given by \[ A(u)=div\{\frac{\nabla_ xu}{(1+| \nabla_ xu|^ 2)^{1/2}}\} \] representing the mean curvature operator. Another choice of A was considered by Galaktionov, \(A(u)=\sum^{N}_{i=1}\partial_{x_ i}(| \nabla u|^{\sigma} \partial_{x_ i}u).\)

Part 1.3 of section 1 contains a summary of results for bounded domains D, while the part 1.4 is devoted to systems of equations. For example \(u_ t=\Delta u+v^ p\), \(v_ t=\Delta v+u^ p.\)

In section II the author considers the nonlinear Schrödinger equation \[ (NLS)\quad iu_ t+\Delta u+| u|^{p-1}u=0,\quad x\in {\mathbb{R}}^ N,\quad t>0,\quad u(0,x)=u_ 0(x). \] For this problem the critical exponent is \(p_{nls}(N)=1+4/N.\)

In section III the nonlinear wave equation \(u_{tt}=\Delta u+| u|^ p\), \(u(0,x)=u_ 0(x)\), \(u_ t(0,x)=u_ 1(x)\) as well as the critical exponent for this equation are examined. The critical exponent is the larger root of the quadratic equation \((N-1)p^ 2-(N+1)p-2=0.\)

Finally, in section IV some concluding remarks are discussed.

Reviewer: V.Georgiev

##### MSC:

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35K55 | Nonlinear parabolic equations |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35K65 | Degenerate parabolic equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35L70 | Second-order nonlinear hyperbolic equations |