In this thesis, we shall give a concise account for the classification of the structure of positive radial solutions of the semilinear elliptic equation$$Delta u+K(|x|)u^{p}=0 .$$ It is known that a radial solution $u$ is crossing if $u$ has a zero in $(0, infty)$; $u$ is slowly decaying if $u$ is positive but $displaystylelim_{r ightarrow{infty}}r^{n-2}u=infty$; u is rapidly decaying if $u$ is positive, $displaystylelim_{r ightarrow{infty}}r^{n-2}u$ exists and is positive. Using some Pohozaev identities, we show that under certain condition on $K$, by comparing some parameters $r_{G}$ and $r_{H}$, the structure of positive radial solutions for various initial conditions can be classified as Type Z ($u(r; alpha)$ is crossing for all $r>0$ ), Type S ($u(r; alpha)$ is slowly decaying for all $r>0$), and Type M (there is some $alpha_{f}$ such that $u(r; alpha)$ is crossing for $alphain(alpha_{f}, infty)$, $u(r; alpha)$ is slowly decaying for $alpha=alpha_{f}$, and $u(r; alpha)$ is rapidly decaying for $alphain(0, alpha_{f})$). The above work is due to Yanagida and Yotsutani.

We prove two results; first, we show that a boundary value problem for the semilinear wave equation with smooth, asymptotically linear nonlinearity and sinusoidal smooth forcing along a characteristic cannot have a continuous solution. Thereafter, we show that if the sinusoidal forcing is not isolated to a characteristic of the wave equation, then the problem has a continuous solution.

The elliptic equation $Delta u+K(|x|)|u|^{p-1}u=0,xin mathbf{R}^{n}$ is studied, where $p>1$, $n>2$, $K(r)$ is smooth and positive on $(0,infty)$, and $rK(r)in L^{1}(0,1)$. It is known that the radial solution either oscillates infinitely, or $lim_{r ightarrow infty}r^{n-2}u(r;al) in Rsetminus {0}$ (rapidly decaying), or $lim_{r ightarrow infty}r^{n-2}u(r;al) = infty (or -infty)$ (slowly decaying). Let $u=u(r;al)$ is a solution satisfying $u(0)=al$. In this thesis, we classify all the radial solutions into three types: Type R($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is rapidly decaying at $r=infty$. Type S($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is slowly decaying at $r=infty$. Type O: $u$ has infinitely many zeros on $(0,infty)$. If $rK_{r}(r)/K(r)$ satisfies some conditions, then the structure of radial solutions is determined completely. In particular, there exists $0<al_{0}<al_{1}<al_{2}<cdots<infty$ such that $u(r;al_{i})$ is of Type R($i$), and $u(r;al)$ is of Type S($i$) for all $al in (al_{i-1},al_{i})$, where $al_{-1}:=0$. These works are due to Yanagida and Yotsutani. Their main tools are Kelvin transformation, Pr"{u}fer transformation, and a Pohozaev identity. Here we give a concise account. Also, I impose a concept so called $r-mu graph$, and give two proofs of the Pohozaev identity.

We prove the existence and nonexistence of radial solutions of singular semilinear equations Δu + k(x)f(u)=0 with boundary condition on the exterior of the ball with radius R>0 in ℝ^N such that lim r →∞ u(r)=0, where f: ℝ \ {0} →ℝ is an odd and locally Lipschitz continuous nonlinear function such that there exists a β >0 with f <0 on (0, β), f >0 on (β, ∞), and K(r) ~ r^-α for some α >0.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-05T13:44:39Z No. of bitstreams: 1 FABRÍCIO LOPES DE ARAÚJO PAZ - DISSERTAÇÃO PPGMAT 2012..pdf: 1112400 bytes, checksum: bb5a2bfc91abd1944b395cfa18b977da (MD5) / Made available in DSpace on 2018-08-05T13:44:39Z (GMT). No. of bitstreams: 1 FABRÍCIO LOPES DE ARAÚJO PAZ - DISSERTAÇÃO PPGMAT 2012..pdf: 1112400 bytes, checksum: bb5a2bfc91abd1944b395cfa18b977da (MD5) Previous issue date: 2012-06 / Capes / Para ler o resumo deste trabalho recomendamos o download do arquivo uma vez que o mesmo possui fórmulas e símbolos matemáticos que não puderam ser transcritos neste espaço. / To read the summary of this work we recommend downloading the file as it contains mathematical formulas and symbols that could not be transcribed in this space.

We prove the existence and nonexistence of solutions for the semilinear problem ∆u + K(r)f(u) = 0 with various boundary conditions on the exterior of the ball in R^N such that lim r→∞u(r) = 0. Here f : R → R is an odd locally lipschitz non-linear function such that there exists a β > 0 with f < 0 on (0, β), f > 0 on (β, ∞), and K(r) \equiv r^−α for some α > 0.

Submitted by MARCO ANTONIO MOYA ROSAS null (23689400813) on 2016-06-08T23:48:22Z No. of bitstreams: 1 Marco.pdf: 3930307 bytes, checksum: fa27a7c6ffca34b6c67e9ba1944a88be (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-06-13T14:55:05Z (GMT) No. of bitstreams: 1 rosas_mm_me_rcla.pdf: 3930307 bytes, checksum: fa27a7c6ffca34b6c67e9ba1944a88be (MD5) / Made available in DSpace on 2016-06-13T14:55:05Z (GMT). No. of bitstreams: 1 rosas_mm_me_rcla.pdf: 3930307 bytes, checksum: fa27a7c6ffca34b6c67e9ba1944a88be (MD5) Previous issue date: 2016-04-28 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho, analisaremos o problema de boa colocação do problema de valor inicial para a equação semilinear do calor. Mostraremos a existência de solução global mild, quando o dado inicial u_0 pertence ao espaço L^( n(ρ−1) /2) −fraco e tem norma suficientemente pequena. / In this work, we discuss the well−posedness of the initial value problem for the semilinear heat equation. We show the existence of global mild solution, when the initial data u_0 belong to weak L^(n(ρ−1)/ 2) space with a sufficiently small norm.

Made available in DSpace on 2014-06-12T18:28:19Z (GMT). No. of bitstreams: 2 arquivo4243_1.pdf: 672080 bytes, checksum: 6deff6a05389ae357d763b61ce1acd73 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2008 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Obtivemos a boa-colocação global de soluções pequenas em espaços de Marcinkiewicz L(p1;¥) £L(p2;¥) para um sistema semilinear. Soluções brandas são obtidas em espaços com índices certos para permitir a existência de solução auto-similar. Usando nossas estimativas dos termos de acoplamento não lineares, provamos a unicidade de soluções na classe C([0;¥);Lp1(Rn)£Lp2(Rn)); sem qualquer hipótese de pequenez. Provamos algumas estimativas de decaimento e analisamos o comportamento assintótico das soluções. Estudamos também o problema de Cauchy para a equação da onda semilinear, com dados singulares em espaços de Marcinkiewicz, provando um resultado de boa-colocação local e decaimento próximo de t = 0

Considere o problema de valor inicial e de fronteira \'u IND.t\'= \'delta\'u + f(u) em \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) se x \'PERTENCE A\' \'ômega\', u(x, t) = 0 se x \'PERTENCE A\' \'delta\' \'ômega\', 0 < t < T, onde ­\'ômega\' é um domínio limitado em \'R POT.n\'com bordo \'C POT.2\', f é continuamente diferenciável com f(s) > 0, e \'fi\' é não-negativa e suave sobre \'ômega\'\'BARRA\' com \'fi\'=0 sobre \'delta\'\'ômega\'. Suponha que a única solução u(x,t) possui blow-up em tempo finito T < \'INFINITO\'. A questão que se coloca é: onde ocorre o blow-up? Neste trabalho provamos que: se \'ômega\'=\'B IND.R\'\'ESTÁ CONTIDO EM\'\'R POT. n\', então o blow-up ocorre apenas em r=0, Além disso, se f(u)=\'u POT.p\'p > 1, então u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) para qualquer 1 < \'gama\'< p, e assim \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. No caso não simétrico onde \'ômega\' é um domínio complexo, provamos que conjunto de blow-up é um subconjunto compacto de \'ômega\'. Se f(u)=\'u POT.p\', p > 1, então u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' e, se n=1,2 ou se n\'< OU=\'3 p\'< OU=\'(n+2)/(n-2), então \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' quando \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'onde \'beta\'= \'(p-1) POT. -1\'. As provas das estimativas essenciais para demonstração desses resultados são feitas utilizando o Princípio do Máximo / Consider the initial-boundary value problem \'u IND.t\'= \'delta\'u + f(u) in \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) if x \'BELONGS\' \'ômega\', u(x, t) = 0 if x \'BELONGS \' \'\\PARTIAL\' \'ômega\', 0 < t < T, where ­\'ômega\' is a bounded domain in \'R POT.n\'with \'C POT.2\', f is continuously differentiable with f(s) > 0, and \'fi\' is nonnegative and smooth on \'ômega\'\'BARRA\' with \'fi\'=0 on \'\\PARTIIAL\'\'ômega\'. Assume that the unique solution u(x,t) blows up in finite time T < \'INFINITO\'. The question addressed is: where does the blow-up occur? In this work we prove: if \'ômega\'=\'B IND.R\'\'IS CONTAINED EM\'\'R POT. n\', then blow-up occurs only at r=0, Moreover, if f(u)=\'u POT.p\'p > 1, then u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) for any 1 < \'gama\'< p, and hence \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. In the nonsymmetric case where \'ômega\' is a convex domain, we prove that the blow-up set lies in a compact subset of \'ômega\'. If f(u)=\'u POT.p\', p > 1, then u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' and, if n=1,2 or if n\'< OU=\'3 and p\'< OU=\'(n+2)/(n-2), then \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' where \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'where \'beta\'= \'(p-1) POT. -1\'. Elementary applications of the Maximum Principle are used to prove the essential estimate for the proofs of these results.

Considere o problema de valor inicial e de fronteira \'u IND.t\'= \'delta\'u + f(u) em \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) se x \'PERTENCE A\' \'ômega\', u(x, t) = 0 se x \'PERTENCE A\' \'delta\' \'ômega\', 0 < t < T, onde ­\'ômega\' é um domínio limitado em \'R POT.n\'com bordo \'C POT.2\', f é continuamente diferenciável com f(s) > 0, e \'fi\' é não-negativa e suave sobre \'ômega\'\'BARRA\' com \'fi\'=0 sobre \'delta\'\'ômega\'. Suponha que a única solução u(x,t) possui blow-up em tempo finito T < \'INFINITO\'. A questão que se coloca é: onde ocorre o blow-up? Neste trabalho provamos que: se \'ômega\'=\'B IND.R\'\'ESTÁ CONTIDO EM\'\'R POT. n\', então o blow-up ocorre apenas em r=0, Além disso, se f(u)=\'u POT.p\'p > 1, então u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) para qualquer 1 < \'gama\'< p, e assim \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. No caso não simétrico onde \'ômega\' é um domínio complexo, provamos que conjunto de blow-up é um subconjunto compacto de \'ômega\'. Se f(u)=\'u POT.p\', p > 1, então u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' e, se n=1,2 ou se n\'< OU=\'3 p\'< OU=\'(n+2)/(n-2), então \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' quando \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'onde \'beta\'= \'(p-1) POT. -1\'. As provas das estimativas essenciais para demonstração desses resultados são feitas utilizando o Princípio do Máximo / Consider the initial-boundary value problem \'u IND.t\'= \'delta\'u + f(u) in \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) if x \'BELONGS\' \'ômega\', u(x, t) = 0 if x \'BELONGS \' \'\\PARTIAL\' \'ômega\', 0 < t < T, where ­\'ômega\' is a bounded domain in \'R POT.n\'with \'C POT.2\', f is continuously differentiable with f(s) > 0, and \'fi\' is nonnegative and smooth on \'ômega\'\'BARRA\' with \'fi\'=0 on \'\\PARTIIAL\'\'ômega\'. Assume that the unique solution u(x,t) blows up in finite time T < \'INFINITO\'. The question addressed is: where does the blow-up occur? In this work we prove: if \'ômega\'=\'B IND.R\'\'IS CONTAINED EM\'\'R POT. n\', then blow-up occurs only at r=0, Moreover, if f(u)=\'u POT.p\'p > 1, then u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) for any 1 < \'gama\'< p, and hence \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. In the nonsymmetric case where \'ômega\' is a convex domain, we prove that the blow-up set lies in a compact subset of \'ômega\'. If f(u)=\'u POT.p\', p > 1, then u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' and, if n=1,2 or if n\'< OU=\'3 and p\'< OU=\'(n+2)/(n-2), then \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' where \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'where \'beta\'= \'(p-1) POT. -1\'. Elementary applications of the Maximum Principle are used to prove the essential estimate for the proofs of these results.